Abstract

Compensation of chromatic dispersion for the optical implementation of mathematical transformations has proved to be an important tool in the design of new optical methods for full-color signal processing. A novel approach for designing dispersion-compensated, broadband optical transformers, both Fourier and Fresnel, based on the collimated Fresnel number is introduced. In a second stage, the above framework is fully exploited to achieve the optical implementation of the fractional Fourier transform (FRT) of any diffracting screen with broadband illumination. Moreover, we demonstrate that the amount of shift variance of the dispersion-compensated FRT can be tuned continuously from the spatial domain, which is totally space variant, to the spectral domain, which is totally space invariant, with the chromatic correction remaining unaltered.

© 2004 Optical Society of America

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  1. 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.
  2. G. P. Behrmann, J. N. Mait, “Hybrid (refractive/diffractive) optics,” in Micro-Optics: Elements, systems and applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 259–292.
  3. G. M. Morris, K. J. McIntyre, “Optical system design with diffractive optics,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie, Berlin, 1997), pp. 81–101.
  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. 24, 3638–3642 (1985).
    [CrossRef] [PubMed]
  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. 31, 4682–4687 (1992).
    [CrossRef] [PubMed]
  12. M. Schwab, N. Lindlein, J. Schwider, Y. Amitai, A. A. Friesem, S. Reinhorn, “Compensation of the wavelength dependence in diffractive star couplers,” J. Opt. Soc. Am. A 12, 1290–1297 (1995).
    [CrossRef]
  13. J. Schwider, “Achromatic design of holographic optical interconnects,” Opt. Eng. 35, 826–831 (1996).
    [CrossRef]
  14. J. Lancis, E. Tajahuerce, P. Andrés, G. Mı́nguez-Vega, M. Fernández-Alonso, V. Climent, “Quasi-wavelength-independent broadband optical Fourier transformer,” Opt. Commun. 172, 153–160 (1999).
    [CrossRef]
  15. D. Wang, A. Pe’er, A. W. Lohmann, A. A. Friesem, “Wigner algebra as a tool for the design of achromatic optical processing systems,” Opt. Eng. 39, 3014–3024 (2000).
    [CrossRef]
  16. J. Lancis, P. Andrés, W. D. Furlan, A. Pons, “All-diffractive achromatic Fourier-transform setup,” Opt. Lett. 19, 402–404 (1994).
    [PubMed]
  17. E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
    [CrossRef]
  18. E. Tajahuerce, P. Andrés, J. Lancis, M. Fernández-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
    [CrossRef]
  19. G. M. Morris, N. George, “Frequency-plane filtering with an achromatic optical transform,” Opt. Lett. 5, 446–448 (1980).
    [CrossRef] [PubMed]
  20. R. Ferrière, C. Illueca, J. P. Goedgebuer, “Corrélateur achromatique bidimensionnel,” J. Opt. (Paris) 17, 153–159 (1986).
  21. E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid #(refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
    [CrossRef]
  22. M. Domingo, I. Arias, A. Garcı́a, “Achromatic Fourier processor with holographic optical lenses,” Appl. Opt. 40, 2267–2274 (2001).
    [CrossRef]
  23. D. Mendlovic, Z. Zalevsky, P. Andrés, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).
  24. K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 3–108 (1989).
  25. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  26. P. Andrés, W. D. Furlan, G. Saavedra, A. W. Lohmann, “Variable fractional Fourier processor: a simple implementation,” J. Opt. Soc. Am. A 14, 853–858 (1997).
    [CrossRef]
  27. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 265–342 (1998).
  28. H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing (Wiley, New York, 2000).
  29. A. Torre, “The fractional Fourier transform and some of its applications to optics,” Prog. Opt. 43, 531–596 (2002).
    [CrossRef]
  30. D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transform to optical pattern recognition,” in Optical Pattern Recognition, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, New York, 1998), pp. 89–125.
  31. J. A. Davis, D. M. Cottrell, N. Nestorovic, S. M. Highnote, “Space-variant Fresnel transform optical correlator,” Appl. Opt. 31, 6889–6893 (1992).
    [CrossRef] [PubMed]
  32. E. E. Sicre, N. Bolognini, M. Garavaglia, “Partial achromatization of the self-imaging phenomenon,” Appl. Opt. 24, 929–930 (1985).
    [CrossRef]
  33. G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
    [CrossRef]
  34. R. H. Katyl, “Compensating optical systems. Part 2: generation of holograms with broadband light,” Appl. Opt. 11, 1248–1254 (1972).
    [CrossRef] [PubMed]
  35. B. Packross, R. Eschbach, O. Bryngdahl, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
    [CrossRef]
  36. P. Andrés, J. Lancis, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
    [CrossRef]
  37. J. Lancis, E. E. Sicre, A. Pons, G. Saavedra, “Achromatic white-light self-imaging phenomenon: an approach using the Wigner distribution function,” J. Mod. Opt. 42, 425–434 (1995).
    [CrossRef]
  38. J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
    [CrossRef]
  39. J. Lancis, G. Mı́nguez-Vega, E. Tajahuerce, M. Fernández-Alonso, V. Climent, P. Andrés, “Wavelength-compensated Fourier and Fresnel transformers: a unified approach,” Opt. Lett. 27, 942–944 (2002).
    [CrossRef]
  40. E. Tajahuerce, G. Saavedra, W. D. Furlan, E. E. Sicre, P. Andrés, “White-light optical implementation of the fractional Fourier transform with adjustable order control,” Appl. Opt. 39, 238–245 (2000).
    [CrossRef]
  41. E. Tajahuerce, E. Bonet, P. Andrés, C. J. Zapata-Rodrı́guez, V. Climent, “White-light modified Talbot array illuminator with a variable density of light spots,” Appl. Opt. 37, 4366–4373 (1998).
    [CrossRef]
  42. E. Tajahuerce, E. Bonet, J. Lancis, M. T. Gale, P. Andrés, “Achromatic fan-out diffractive system for white-light free-space optical interconnects,” J. Mod. Opt. 48, 831–845 (2001).
    [CrossRef]
  43. G. Mı́nguez-Vega, J. Lancis, E. Tajahuerce, V. Climent, J. Caraquitena, P. Andrés, “Broadband space-variant Fresnel processor,” Opt. Lett. 27, 1926–1929 (2002).
    [CrossRef]
  44. J. Lancis, G. Mı́nguez-Vega, E. Tajahuerce, V. Climent, P. Andrés, Z. Jaroszewicz, “High-contrast white-light Lau fringes,” Opt. Lett. 29, 150–152 (2004).
    [CrossRef] [PubMed]
  45. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  46. A. Yariv, “Imaging of coherent fields through lenslike systems,” Opt. Lett. 19, 1607–1608 (1994).
    [CrossRef] [PubMed]

2004

2002

2001

M. Domingo, I. Arias, A. Garcı́a, “Achromatic Fourier processor with holographic optical lenses,” Appl. Opt. 40, 2267–2274 (2001).
[CrossRef]

E. Tajahuerce, E. Bonet, J. Lancis, M. T. Gale, P. Andrés, “Achromatic fan-out diffractive system for white-light free-space optical interconnects,” J. Mod. Opt. 48, 831–845 (2001).
[CrossRef]

2000

E. Tajahuerce, G. Saavedra, W. D. Furlan, E. E. Sicre, P. Andrés, “White-light optical implementation of the fractional Fourier transform with adjustable order control,” Appl. Opt. 39, 238–245 (2000).
[CrossRef]

D. Wang, A. Pe’er, A. W. Lohmann, A. A. Friesem, “Wigner algebra as a tool for the design of achromatic optical processing systems,” Opt. Eng. 39, 3014–3024 (2000).
[CrossRef]

1999

J. Lancis, E. Tajahuerce, P. Andrés, G. Mı́nguez-Vega, M. Fernández-Alonso, V. Climent, “Quasi-wavelength-independent broadband optical Fourier transformer,” Opt. Commun. 172, 153–160 (1999).
[CrossRef]

E. Tajahuerce, P. Andrés, J. Lancis, M. Fernández-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[CrossRef]

D. Mendlovic, Z. Zalevsky, P. Andrés, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

1998

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid #(refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

E. Tajahuerce, E. Bonet, P. Andrés, C. J. Zapata-Rodrı́guez, V. Climent, “White-light modified Talbot array illuminator with a variable density of light spots,” Appl. Opt. 37, 4366–4373 (1998).
[CrossRef]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 265–342 (1998).

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
[CrossRef]

1997

P. Andrés, W. D. Furlan, G. Saavedra, A. W. Lohmann, “Variable fractional Fourier processor: a simple implementation,” J. Opt. Soc. Am. A 14, 853–858 (1997).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

1996

J. Schwider, “Achromatic design of holographic optical interconnects,” Opt. Eng. 35, 826–831 (1996).
[CrossRef]

1995

M. Schwab, N. Lindlein, J. Schwider, Y. Amitai, A. A. Friesem, S. Reinhorn, “Compensation of the wavelength dependence in diffractive star couplers,” J. Opt. Soc. Am. A 12, 1290–1297 (1995).
[CrossRef]

J. Lancis, E. E. Sicre, A. Pons, G. Saavedra, “Achromatic white-light self-imaging phenomenon: an approach using the Wigner distribution function,” J. Mod. Opt. 42, 425–434 (1995).
[CrossRef]

1994

1993

P. Andrés, J. Lancis, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
[CrossRef]

1992

1989

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 3–108 (1989).

1988

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[CrossRef]

1986

R. Ferrière, C. Illueca, J. P. Goedgebuer, “Corrélateur achromatique bidimensionnel,” J. Opt. (Paris) 17, 153–159 (1986).

1985

1984

B. Packross, R. Eschbach, O. Bryngdahl, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[CrossRef]

1983

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

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

1981

1980

1979

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

1972

Alper Kutay, M.

H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Amitai, Y.

Andrés, P.

J. Lancis, G. Mı́nguez-Vega, E. Tajahuerce, V. Climent, P. Andrés, Z. Jaroszewicz, “High-contrast white-light Lau fringes,” Opt. Lett. 29, 150–152 (2004).
[CrossRef] [PubMed]

G. Mı́nguez-Vega, J. Lancis, E. Tajahuerce, V. Climent, J. Caraquitena, P. Andrés, “Broadband space-variant Fresnel processor,” Opt. Lett. 27, 1926–1929 (2002).
[CrossRef]

J. Lancis, G. Mı́nguez-Vega, E. Tajahuerce, M. Fernández-Alonso, V. Climent, P. Andrés, “Wavelength-compensated Fourier and Fresnel transformers: a unified approach,” Opt. Lett. 27, 942–944 (2002).
[CrossRef]

E. Tajahuerce, E. Bonet, J. Lancis, M. T. Gale, P. Andrés, “Achromatic fan-out diffractive system for white-light free-space optical interconnects,” J. Mod. Opt. 48, 831–845 (2001).
[CrossRef]

E. Tajahuerce, G. Saavedra, W. D. Furlan, E. E. Sicre, P. Andrés, “White-light optical implementation of the fractional Fourier transform with adjustable order control,” Appl. Opt. 39, 238–245 (2000).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, G. Mı́nguez-Vega, M. Fernández-Alonso, V. Climent, “Quasi-wavelength-independent broadband optical Fourier transformer,” Opt. Commun. 172, 153–160 (1999).
[CrossRef]

E. Tajahuerce, P. Andrés, J. Lancis, M. Fernández-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[CrossRef]

D. Mendlovic, Z. Zalevsky, P. Andrés, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid #(refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
[CrossRef]

E. Tajahuerce, E. Bonet, P. Andrés, C. J. Zapata-Rodrı́guez, V. Climent, “White-light modified Talbot array illuminator with a variable density of light spots,” Appl. Opt. 37, 4366–4373 (1998).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

P. Andrés, W. D. Furlan, G. Saavedra, A. W. Lohmann, “Variable fractional Fourier processor: a simple implementation,” J. Opt. Soc. Am. A 14, 853–858 (1997).
[CrossRef]

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, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
[CrossRef]

P. Andrés, J. Lancis, W. D. Furlan, “White-light Fourier transformer with low chromatic aberration,” Appl. Opt. 31, 4682–4687 (1992).
[CrossRef] [PubMed]

Arias, I.

Behrmann, G. P.

G. P. Behrmann, J. N. Mait, “Hybrid (refractive/diffractive) optics,” in Micro-Optics: Elements, systems and applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 259–292.

Bolognini, N.

Bonet, E.

E. Tajahuerce, E. Bonet, J. Lancis, M. T. Gale, P. Andrés, “Achromatic fan-out diffractive system for white-light free-space optical interconnects,” J. Mod. Opt. 48, 831–845 (2001).
[CrossRef]

E. Tajahuerce, E. Bonet, P. Andrés, C. J. Zapata-Rodrı́guez, V. Climent, “White-light modified Talbot array illuminator with a variable density of light spots,” Appl. Opt. 37, 4366–4373 (1998).
[CrossRef]

P. Andrés, J. Lancis, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
[CrossRef]

Brophy, C.

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

Bryngdahl, O.

B. Packross, R. Eschbach, O. Bryngdahl, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[CrossRef]

Caraquitena, J.

Climent, V.

J. Lancis, G. Mı́nguez-Vega, E. Tajahuerce, V. Climent, P. Andrés, Z. Jaroszewicz, “High-contrast white-light Lau fringes,” Opt. Lett. 29, 150–152 (2004).
[CrossRef] [PubMed]

J. Lancis, G. Mı́nguez-Vega, E. Tajahuerce, M. Fernández-Alonso, V. Climent, P. Andrés, “Wavelength-compensated Fourier and Fresnel transformers: a unified approach,” Opt. Lett. 27, 942–944 (2002).
[CrossRef]

G. Mı́nguez-Vega, J. Lancis, E. Tajahuerce, V. Climent, J. Caraquitena, P. Andrés, “Broadband space-variant Fresnel processor,” Opt. Lett. 27, 1926–1929 (2002).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, G. Mı́nguez-Vega, M. Fernández-Alonso, V. Climent, “Quasi-wavelength-independent broadband optical Fourier transformer,” Opt. Commun. 172, 153–160 (1999).
[CrossRef]

E. Tajahuerce, P. Andrés, J. Lancis, M. Fernández-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[CrossRef]

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
[CrossRef]

E. Tajahuerce, E. Bonet, P. Andrés, C. J. Zapata-Rodrı́guez, V. Climent, “White-light modified Talbot array illuminator with a variable density of light spots,” Appl. Opt. 37, 4366–4373 (1998).
[CrossRef]

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid #(refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

Cottrell, D. M.

Davis, J. A.

Domingo, M.

Eschbach, R.

B. Packross, R. Eschbach, O. Bryngdahl, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[CrossRef]

Fernández-Alonso, M.

J. Lancis, G. Mı́nguez-Vega, E. Tajahuerce, M. Fernández-Alonso, V. Climent, P. Andrés, “Wavelength-compensated Fourier and Fresnel transformers: a unified approach,” Opt. Lett. 27, 942–944 (2002).
[CrossRef]

E. Tajahuerce, P. Andrés, J. Lancis, M. Fernández-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, G. Mı́nguez-Vega, M. Fernández-Alonso, V. Climent, “Quasi-wavelength-independent broadband optical Fourier transformer,” Opt. Commun. 172, 153–160 (1999).
[CrossRef]

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
[CrossRef]

Ferrière, R.

R. Ferrière, C. Illueca, J. P. Goedgebuer, “Corrélateur achromatique bidimensionnel,” J. Opt. (Paris) 17, 153–159 (1986).

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

Friesem, A. A.

D. Wang, A. Pe’er, A. W. Lohmann, A. A. Friesem, “Wigner algebra as a tool for the design of achromatic optical processing systems,” Opt. Eng. 39, 3014–3024 (2000).
[CrossRef]

M. Schwab, N. Lindlein, J. Schwider, Y. Amitai, A. A. Friesem, S. Reinhorn, “Compensation of the wavelength dependence in diffractive star couplers,” J. Opt. Soc. Am. A 12, 1290–1297 (1995).
[CrossRef]

Furlan, W. D.

Gale, M. T.

E. Tajahuerce, E. Bonet, J. Lancis, M. T. Gale, P. Andrés, “Achromatic fan-out diffractive system for white-light free-space optical interconnects,” J. Mod. Opt. 48, 831–845 (2001).
[CrossRef]

Garavaglia, M.

Garci´a, A.

George, N.

Goedgebuer, J. P.

R. Ferrière, C. Illueca, J. P. Goedgebuer, “Corrélateur achromatique bidimensionnel,” J. Opt. (Paris) 17, 153–159 (1986).

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

Highnote, S. M.

Illueca, C.

R. Ferrière, C. Illueca, J. P. Goedgebuer, “Corrélateur achromatique bidimensionnel,” J. Opt. (Paris) 17, 153–159 (1986).

Indebetouw, G.

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[CrossRef]

Jaroszewicz, Z.

Katyl, R. H.

Lancis, J.

J. Lancis, G. Mı́nguez-Vega, E. Tajahuerce, V. Climent, P. Andrés, Z. Jaroszewicz, “High-contrast white-light Lau fringes,” Opt. Lett. 29, 150–152 (2004).
[CrossRef] [PubMed]

G. Mı́nguez-Vega, J. Lancis, E. Tajahuerce, V. Climent, J. Caraquitena, P. Andrés, “Broadband space-variant Fresnel processor,” Opt. Lett. 27, 1926–1929 (2002).
[CrossRef]

J. Lancis, G. Mı́nguez-Vega, E. Tajahuerce, M. Fernández-Alonso, V. Climent, P. Andrés, “Wavelength-compensated Fourier and Fresnel transformers: a unified approach,” Opt. Lett. 27, 942–944 (2002).
[CrossRef]

E. Tajahuerce, E. Bonet, J. Lancis, M. T. Gale, P. Andrés, “Achromatic fan-out diffractive system for white-light free-space optical interconnects,” J. Mod. Opt. 48, 831–845 (2001).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, G. Mı́nguez-Vega, M. Fernández-Alonso, V. Climent, “Quasi-wavelength-independent broadband optical Fourier transformer,” Opt. Commun. 172, 153–160 (1999).
[CrossRef]

E. Tajahuerce, P. Andrés, J. Lancis, M. Fernández-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[CrossRef]

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
[CrossRef]

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid #(refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

J. Lancis, E. E. Sicre, A. Pons, G. Saavedra, “Achromatic white-light self-imaging phenomenon: an approach using the Wigner distribution function,” J. Mod. Opt. 42, 425–434 (1995).
[CrossRef]

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, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
[CrossRef]

P. Andrés, J. Lancis, W. D. Furlan, “White-light Fourier transformer with low chromatic aberration,” Appl. Opt. 31, 4682–4687 (1992).
[CrossRef] [PubMed]

Leith, E. N.

Leon, S.

Lindlein, N.

Lohmann, A. W.

D. Wang, A. Pe’er, A. W. Lohmann, A. A. Friesem, “Wigner algebra as a tool for the design of achromatic optical processing systems,” Opt. Eng. 39, 3014–3024 (2000).
[CrossRef]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 265–342 (1998).

P. Andrés, W. D. Furlan, G. Saavedra, A. W. Lohmann, “Variable fractional Fourier processor: a simple implementation,” J. Opt. Soc. Am. A 14, 853–858 (1997).
[CrossRef]

Mait, J. N.

G. P. Behrmann, J. N. Mait, “Hybrid (refractive/diffractive) optics,” in Micro-Optics: Elements, systems and applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 259–292.

McIntyre, K. J.

G. M. Morris, K. J. McIntyre, “Optical system design with diffractive optics,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie, Berlin, 1997), pp. 81–101.

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, P. Andrés, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 265–342 (1998).

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transform to optical pattern recognition,” in Optical Pattern Recognition, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, New York, 1998), pp. 89–125.

Mi´nguez-Vega, G.

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, N. George, “Frequency-plane filtering with an achromatic optical transform,” Opt. Lett. 5, 446–448 (1980).
[CrossRef] [PubMed]

G. M. Morris, K. J. McIntyre, “Optical system design with diffractive optics,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie, Berlin, 1997), pp. 81–101.

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.

Nestorovic, N.

Ozaktas, H. M.

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transform to optical pattern recognition,” in Optical Pattern Recognition, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, New York, 1998), pp. 89–125.

H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Packross, B.

B. Packross, R. Eschbach, O. Bryngdahl, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[CrossRef]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 3–108 (1989).

Pe’er, A.

D. Wang, A. Pe’er, A. W. Lohmann, A. A. Friesem, “Wigner algebra as a tool for the design of achromatic optical processing systems,” Opt. Eng. 39, 3014–3024 (2000).
[CrossRef]

Pellat-Finet, P.

Pons, A.

J. Lancis, E. E. Sicre, A. Pons, G. Saavedra, “Achromatic white-light self-imaging phenomenon: an approach using the Wigner distribution function,” J. Mod. Opt. 42, 425–434 (1995).
[CrossRef]

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

Reinhorn, S.

Saavedra, G.

Schwab, M.

Schwider, J.

Sicre, E. E.

E. Tajahuerce, G. Saavedra, W. D. Furlan, E. E. Sicre, P. Andrés, “White-light optical implementation of the fractional Fourier transform with adjustable order control,” Appl. Opt. 39, 238–245 (2000).
[CrossRef]

J. Lancis, E. E. Sicre, A. Pons, G. Saavedra, “Achromatic white-light self-imaging phenomenon: an approach using the Wigner distribution function,” J. Mod. Opt. 42, 425–434 (1995).
[CrossRef]

P. Andrés, J. Lancis, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
[CrossRef]

E. E. Sicre, N. Bolognini, M. Garavaglia, “Partial achromatization of the self-imaging phenomenon,” Appl. Opt. 24, 929–930 (1985).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Tajahuerce, E.

J. Lancis, G. Mı́nguez-Vega, E. Tajahuerce, V. Climent, P. Andrés, Z. Jaroszewicz, “High-contrast white-light Lau fringes,” Opt. Lett. 29, 150–152 (2004).
[CrossRef] [PubMed]

G. Mı́nguez-Vega, J. Lancis, E. Tajahuerce, V. Climent, J. Caraquitena, P. Andrés, “Broadband space-variant Fresnel processor,” Opt. Lett. 27, 1926–1929 (2002).
[CrossRef]

J. Lancis, G. Mı́nguez-Vega, E. Tajahuerce, M. Fernández-Alonso, V. Climent, P. Andrés, “Wavelength-compensated Fourier and Fresnel transformers: a unified approach,” Opt. Lett. 27, 942–944 (2002).
[CrossRef]

E. Tajahuerce, E. Bonet, J. Lancis, M. T. Gale, P. Andrés, “Achromatic fan-out diffractive system for white-light free-space optical interconnects,” J. Mod. Opt. 48, 831–845 (2001).
[CrossRef]

E. Tajahuerce, G. Saavedra, W. D. Furlan, E. E. Sicre, P. Andrés, “White-light optical implementation of the fractional Fourier transform with adjustable order control,” Appl. Opt. 39, 238–245 (2000).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, G. Mı́nguez-Vega, M. Fernández-Alonso, V. Climent, “Quasi-wavelength-independent broadband optical Fourier transformer,” Opt. Commun. 172, 153–160 (1999).
[CrossRef]

E. Tajahuerce, P. Andrés, J. Lancis, M. Fernández-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[CrossRef]

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
[CrossRef]

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid #(refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

E. Tajahuerce, E. Bonet, P. Andrés, C. J. Zapata-Rodrı́guez, V. Climent, “White-light modified Talbot array illuminator with a variable density of light spots,” Appl. Opt. 37, 4366–4373 (1998).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

Tepichin, E.

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

Torre, A.

A. Torre, “The fractional Fourier transform and some of its applications to optics,” Prog. Opt. 43, 531–596 (2002).
[CrossRef]

Wang, D.

D. Wang, A. Pe’er, A. W. Lohmann, A. A. Friesem, “Wigner algebra as a tool for the design of achromatic optical processing systems,” Opt. Eng. 39, 3014–3024 (2000).
[CrossRef]

Wyne, C. G.

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

Yariv, A.

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, P. Andrés, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 265–342 (1998).

H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing (Wiley, New York, 2000).

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transform to optical pattern recognition,” in Optical Pattern Recognition, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, New York, 1998), pp. 89–125.

Zapata-Rodri´guez, C. J.

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.

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

R. H. Katyl, “Compensating optical systems. Part 3: achromatic Fourier transformation,” Appl. Opt. 11, 1255–1260 (1972).
[CrossRef] [PubMed]

S. Leon, E. N. Leith, “Optical processing and holography with polychromatic point source illumination,” Appl. Opt. 24, 3638–3642 (1985).
[CrossRef] [PubMed]

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

P. Andrés, J. Lancis, W. D. Furlan, “White-light Fourier transformer with low chromatic aberration,” Appl. Opt. 31, 4682–4687 (1992).
[CrossRef] [PubMed]

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
[CrossRef]

M. Domingo, I. Arias, A. Garcı́a, “Achromatic Fourier processor with holographic optical lenses,” Appl. Opt. 40, 2267–2274 (2001).
[CrossRef]

J. A. Davis, D. M. Cottrell, N. Nestorovic, S. M. Highnote, “Space-variant Fresnel transform optical correlator,” Appl. Opt. 31, 6889–6893 (1992).
[CrossRef] [PubMed]

E. E. Sicre, N. Bolognini, M. Garavaglia, “Partial achromatization of the self-imaging phenomenon,” Appl. Opt. 24, 929–930 (1985).
[CrossRef]

R. H. Katyl, “Compensating optical systems. Part 2: generation of holograms with broadband light,” Appl. Opt. 11, 1248–1254 (1972).
[CrossRef] [PubMed]

E. Tajahuerce, G. Saavedra, W. D. Furlan, E. E. Sicre, P. Andrés, “White-light optical implementation of the fractional Fourier transform with adjustable order control,” Appl. Opt. 39, 238–245 (2000).
[CrossRef]

E. Tajahuerce, E. Bonet, P. Andrés, C. J. Zapata-Rodrı́guez, V. Climent, “White-light modified Talbot array illuminator with a variable density of light spots,” Appl. Opt. 37, 4366–4373 (1998).
[CrossRef]

J. Mod. Opt.

E. Tajahuerce, E. Bonet, J. Lancis, M. T. Gale, P. Andrés, “Achromatic fan-out diffractive system for white-light free-space optical interconnects,” J. Mod. Opt. 48, 831–845 (2001).
[CrossRef]

J. Lancis, E. E. Sicre, A. Pons, G. Saavedra, “Achromatic white-light self-imaging phenomenon: an approach using the Wigner distribution function,” J. Mod. Opt. 42, 425–434 (1995).
[CrossRef]

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[CrossRef]

E. Tajahuerce, P. Andrés, J. Lancis, M. Fernández-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[CrossRef]

J. Opt. (Paris)

R. Ferrière, C. Illueca, J. P. Goedgebuer, “Corrélateur achromatique bidimensionnel,” J. Opt. (Paris) 17, 153–159 (1986).

J. Opt. Soc. Am. A

Opt. Commun.

B. Packross, R. Eschbach, O. Bryngdahl, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[CrossRef]

P. Andrés, J. Lancis, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

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]

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid #(refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, G. Mı́nguez-Vega, M. Fernández-Alonso, V. Climent, “Quasi-wavelength-independent broadband optical Fourier transformer,” Opt. Commun. 172, 153–160 (1999).
[CrossRef]

Opt. Eng.

D. Wang, A. Pe’er, A. W. Lohmann, A. A. Friesem, “Wigner algebra as a tool for the design of achromatic optical processing systems,” Opt. Eng. 39, 3014–3024 (2000).
[CrossRef]

J. Schwider, “Achromatic design of holographic optical interconnects,” Opt. Eng. 35, 826–831 (1996).
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

D. Mendlovic, Z. Zalevsky, P. Andrés, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

Prog. Opt.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 3–108 (1989).

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 265–342 (1998).

A. Torre, “The fractional Fourier transform and some of its applications to optics,” Prog. Opt. 43, 531–596 (2002).
[CrossRef]

Other

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transform to optical pattern recognition,” in Optical Pattern Recognition, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, New York, 1998), pp. 89–125.

H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing (Wiley, New York, 2000).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

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.

G. P. Behrmann, J. N. Mait, “Hybrid (refractive/diffractive) optics,” in Micro-Optics: Elements, systems and applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 259–292.

G. M. Morris, K. J. McIntyre, “Optical system design with diffractive optics,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie, Berlin, 1997), pp. 81–101.

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Figures (8)

Fig. 1
Fig. 1

Under white-light, point-source illumination, the Fresnel diffraction pattern achieved at the output plane by free-space propagation is chromatically distorted because of the dependence of the collimated Fresnel number on the wavelength.

Fig. 2
Fig. 2

Hybrid (diffractive-refractive) lens configuration producing a quasi-wavelength-independent Fresnel irradiance profile. See text for definitions. Although the axial distance d is negative in this configuration, a real Fraunhofer plane can be obtained by proper choice of the axial location of S.

Fig. 3
Fig. 3

SCE associated with the Fresnel irradiance profile achieved at the output plane of the novel optical arrangement proposed in Fig. 2 (solid curve). For comparison, the chromatic error CE associated with the normalized irradiance profile achieved at the output plane of the free-space diffraction geometry in Fig. 1 is shown by dashed curve in the same plot. In both cases we assume white-light illumination.

Fig. 4
Fig. 4

Graphical representation of the maximum value of the chromatic error versus the spectral bandwidth Δσ of the illuminating source for (a) the optical arrangement in Fig. 1 (solid curve), (b) free-space diffraction (dashed curve). The design wave number was σ0=1.75 μm-1 in both cases.

Fig. 5
Fig. 5

Black-and-white representation of the irradiance distribution achieved under plane wave illumination at a distance 150.80 mm from the 1D diffraction grating for the monochromatic components of wavelength (a) 488 nm, (b) 568 nm, (c) 647 nm. We recognize that only the Fresnel irradiance profile in Fig. 5(b) corresponds to the self-image of order n=1.5 because of the wavelength dependence of the collimated Fresnel number.

Fig. 6
Fig. 6

Gray-scale picture of the dispersion-compensated self-image of order n=1.5 achieved at the output plane of the optical setup in Fig. 2 for the monochromatic components of wavelength (a) 488 nm, (b) 568 nm, (c) 647 nm.

Fig. 7
Fig. 7

Gray-scale picture of the dispersion-compensated Fraunhofer diffraction pattern of a 1D diffraction grating for the monochromatic components of wavelength (a) 488 nm, (b) 568 nm, (c) 647 nm. This irradiance distribution is achieved at the output plane of the optical setup in Fig. 2 when the diffractive lens DL2 is eliminated. Note that the transversal locations of the diffraction maxima are nearly wavelength independent.

Fig. 8
Fig. 8

Black-and-white representation of the Fraunhofer diffraction pattern of a 1D diffraction grating provided by a conventional, refractive, Fourier-transform objective for the monochromatic components of wavelength (a) 488 nm, (b) 568 nm, (c) 647 nm. Note the strong dependence of the transversal location of the diffraction maxima on the wavelength and that the transversal locations of the diffraction maxima are clearly wavelength dependent.

Equations (51)

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

Uout(r)=-K(r, r0)Uin(r0)d2r0,
K(r, r0)
=σjBexpj πσB (A|r0|2+D|r|2-2rr0),B01Aexpj πσCA |r|2δr0-rA,B=0,
ABCD
w0=r0/s,w=r/As,
U^in(w0)=Uin(r0),
U^out(w)=A exp-j πσCA |r|2Uout(r).
U^out(w)=-jNc-1+1U^in(w0)×exp(jπNc|w-w0|2)d2w0,
NC=Aσs2/B.
Leq=B/A.
Meq=A.
feq=-A/C,
ABCD=10C/A1 A001/A 1B/A01,
U^out(w)=-U^in(w0)exp(-i2πww0)d2w0.
w0=r0/s,w=sσr/B,
U^in(w)=Uin(r0),
U^out(w)=exp-iπ DσB |r|2Uout(r).
GP(w)=FP[g(w0)](w)=i exp(iϕ)sin ϕexpiπtan ϕ |w|2×-g(w0)expiπtan ϕ |w0|2×exp-i2πww0sin ϕd2w0,
U^out(w)=expiπ sin 2ϕ2 |w|2FP[U^in(w0)]wcos ϕ,
P=2πarccot NC.
Uout(r)=expiπ sin 2ϕ2|r|2(As)2FP[U^in(w0)]rAscos ϕ.
ABCD=1R01 101/z1=(z+R)/zR1/z1.
NC=z+RzR s2σ,Meq=z+Rz
NC=s2(z+R)σ0zRσσ0=NC0σσ0,
NC0=s2(z+R)σ0zR
CE(σ)=100 NC-NC0NC0=100 σ-σ0σ0.
CEM=CE(σ2)-CE(σ1).
CEM=100 Δσσ0,
-z(σ)=Z0σσ0-σZ0a-σ0Z0σ-1d-1-1.
ABCD=1d01 10-σ0/Z0σ1 1l01 10-1/f1×1l-a01 101/z1.
NC=s2-σ0Z0+K1σ3+K2σ2+K3σσB(σ)[adσ0-(a+d)Z0σ],
K1=-Z0(l+d)1-l+df+d+l,
K2=σ0ld1-l+df+dd+dl+dZ0Z0(d+l)1-lf+l,
K3=-ddσ02Z0l1-lf+l.
K1K2K30.
l=fl(l-f),Z0=-f2Z0(l-f)2,
d=-d f2(l-f)(d+l-f).
1d+l+1d+l=1f.
NC=-s2σ0Z0,
Leq=-Z0.
Meq(σ)=Z0σ(a+d)-aσ0d(d-f+l)σ2fσ0Z0Z0.
σ Meq(σ)σ0=0.
a=dZ0(2d-Z0),
Meq(σ)=M0(2σ-σ0)σ0σ2,
M0=Meq(σ0)=βSdaZ0Z0.
SCE(σ)=100 M0-Meq(σ)M0
SCE(σ)=100 (σ-σ0)2σ2.
σ0=2 σ1σ2(σ1+σ2).
SCEM=1001+2 1-[1+(Δσ/σ0)2]1/2(Δσ/σ0)2.
P=2πarccot-s2σ0Z0.
Leq(n)=2np2σ,

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