Abstract

We report a diffractive-lens triplet with which to achieve wavelength compensation in the near field diffracted by any aperture. On the one hand, the all-diffractive triplet allows us to tune, in a sequential way, the Fresnel-irradiance shape to be achromatized by changing the focal length of one diffractive lens. On the other hand, we can adjust the scale of the chromatically compensated Fresnel diffraction field by shifting the aperture along the optical axis. Within this framework, we present an extremely flexible white-light Fresnel-plane array illuminator based on the kinoform sampling filter. A variable compression ratio and continuous selection of the output pitch are the most appealing features of this novel application.

© 2005 Optical Society of America

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References

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  1. J. M. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  2. M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).
    [CrossRef]
  3. G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed.(Academic, 1987), Chap. 1.2.
  4. 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]
  5. J. Lancis, E. Tajahuerce, P. Andrés, G. Mínguez-Vega, M. Fernández-Alonso, V. Climent, “Quasi-wavelength-independent broadband optical Fourier transformer,” Opt. Commun. 172, 153–160 (1999).
    [CrossRef]
  6. D. Y. 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]
  7. M. Domingo, I. Arias, A. García, “Achromatic Fourier processor with holographic optical lenses,” Appl. Opt. 40, 2267–2274 (2001).
    [CrossRef]
  8. 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]
  9. 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]
  10. J. Lancis, G. Mínguez-Vega, E. Tajahuerce, V. Climent, P. Andrés, J. Caraquitena, “Chromatic compensation of broadband light diffraction: ABCD-matrix approach,” J. Opt. Soc. Am. A 21, 1875–1885 (2004).
    [CrossRef]
  11. J. Lancis, G. Mínguez-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]
  12. 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 Verlag, 1997), pp. 81–104.
  13. G. Mínguez-Vega, J. Lancis, E. Tajahuerce, V. Climent, J. Caraquitena, P. Andrés, “Broadband space-variant Fresnel processor,” Opt. Lett. 27, 1926–1928 (2002).
    [CrossRef]
  14. S. Sinzinger, J. Jahns, Microoptics (Wiley-VCH, 2003).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  19. A. Kolodziejczyk, Z. Jaroszewicz, A. Kowalik, O. Quintero, “Kinoform sampling filter,” Opt. Commun. 200, 35–42 (2001).
    [CrossRef]
  20. A. Kolodziejczyk, Z. Jaroszewicz, R. Henao, O. Quintero, “The Talbot array illuminator: imaging properties and a new interpretation,” J. Opt. A Pure Appl. Opt. 6, 651–657 (2004).
    [CrossRef]
  21. E. Tajahuerce, E. Bonet, P. Andrés, C. J. Zapata-Rodríguez, V. Climent, “White-light-modified Talbot array illuminator with a variable density of light spots,” Appl. Opt. 37, 4366–4373 (1998).
    [CrossRef]
  22. 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]
  23. N. Guérineau, B. Harchaoui, J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199–203 (2000).
    [CrossRef]
  24. N. Guérineau, B. Harchaoui, J. Primot, K. Heggarty, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. Lett. 26, 411–423 (2001).
    [CrossRef]
  25. A. E. Siegman, Lasers (University Science, 1986).

2004 (2)

J. Lancis, G. Mínguez-Vega, E. Tajahuerce, V. Climent, P. Andrés, J. Caraquitena, “Chromatic compensation of broadband light diffraction: ABCD-matrix approach,” J. Opt. Soc. Am. A 21, 1875–1885 (2004).
[CrossRef]

A. Kolodziejczyk, Z. Jaroszewicz, R. Henao, O. Quintero, “The Talbot array illuminator: imaging properties and a new interpretation,” J. Opt. A Pure Appl. Opt. 6, 651–657 (2004).
[CrossRef]

2002 (2)

2001 (4)

M. Domingo, I. Arias, A. García, “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]

A. Kolodziejczyk, Z. Jaroszewicz, A. Kowalik, O. Quintero, “Kinoform sampling filter,” Opt. Commun. 200, 35–42 (2001).
[CrossRef]

N. Guérineau, B. Harchaoui, J. Primot, K. Heggarty, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. Lett. 26, 411–423 (2001).
[CrossRef]

2000 (3)

N. Guérineau, B. Harchaoui, J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199–203 (2000).
[CrossRef]

D. Y. 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]

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]

1999 (1)

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

1998 (2)

1997 (2)

H. Hamam, “Talbot array illuminators: a general approach,” Appl. Opt. 36, 2319–2327 (1997).
[CrossRef] [PubMed]

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]

1993 (1)

1990 (2)

Andrés, P.

J. Lancis, G. Mínguez-Vega, E. Tajahuerce, V. Climent, P. Andrés, J. Caraquitena, “Chromatic compensation of broadband light diffraction: ABCD-matrix approach,” J. Opt. Soc. Am. A 21, 1875–1885 (2004).
[CrossRef]

J. Lancis, G. Mínguez-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. Mínguez-Vega, J. Lancis, E. Tajahuerce, V. Climent, J. Caraquitena, P. Andrés, “Broadband space-variant Fresnel processor,” Opt. Lett. 27, 1926–1928 (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. Mínguez-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]

E. Tajahuerce, E. Bonet, P. Andrés, C. J. Zapata-Rodríguez, 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]

Arias, I.

Arrizon, V.

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-Rodríguez, V. Climent, “White-light-modified Talbot array illuminator with a variable density of light spots,” Appl. Opt. 37, 4366–4373 (1998).
[CrossRef]

Caraquitena, J.

Climent, V.

Domingo, M.

Fernández-Alonso, M.

Friesem, A. A.

D. Y. 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]

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]

García, A.

Goodman, J. M.

J. M. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gu, M.

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).
[CrossRef]

Guérineau, N.

N. Guérineau, B. Harchaoui, J. Primot, K. Heggarty, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. Lett. 26, 411–423 (2001).
[CrossRef]

N. Guérineau, B. Harchaoui, J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199–203 (2000).
[CrossRef]

Hamam, H.

Harchaoui, B.

N. Guérineau, B. Harchaoui, J. Primot, K. Heggarty, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. Lett. 26, 411–423 (2001).
[CrossRef]

N. Guérineau, B. Harchaoui, J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199–203 (2000).
[CrossRef]

Heggarty, K.

Henao, R.

A. Kolodziejczyk, Z. Jaroszewicz, R. Henao, O. Quintero, “The Talbot array illuminator: imaging properties and a new interpretation,” J. Opt. A Pure Appl. Opt. 6, 651–657 (2004).
[CrossRef]

Jahns, J.

S. Sinzinger, J. Jahns, Microoptics (Wiley-VCH, 2003).
[CrossRef]

Jaroszewicz, Z.

A. Kolodziejczyk, Z. Jaroszewicz, R. Henao, O. Quintero, “The Talbot array illuminator: imaging properties and a new interpretation,” J. Opt. A Pure Appl. Opt. 6, 651–657 (2004).
[CrossRef]

A. Kolodziejczyk, Z. Jaroszewicz, A. Kowalik, O. Quintero, “Kinoform sampling filter,” Opt. Commun. 200, 35–42 (2001).
[CrossRef]

Kolodziejczyk, A.

A. Kolodziejczyk, Z. Jaroszewicz, R. Henao, O. Quintero, “The Talbot array illuminator: imaging properties and a new interpretation,” J. Opt. A Pure Appl. Opt. 6, 651–657 (2004).
[CrossRef]

A. Kolodziejczyk, Z. Jaroszewicz, A. Kowalik, O. Quintero, “Kinoform sampling filter,” Opt. Commun. 200, 35–42 (2001).
[CrossRef]

Kowalik, A.

A. Kolodziejczyk, Z. Jaroszewicz, A. Kowalik, O. Quintero, “Kinoform sampling filter,” Opt. Commun. 200, 35–42 (2001).
[CrossRef]

Lancis, J.

Leger, J. R.

Lohmann, A. W.

D. Y. 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, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
[CrossRef] [PubMed]

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 Verlag, 1997), pp. 81–104.

Mínguez-Vega, G.

Morris, G. M.

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed.(Academic, 1987), Chap. 1.2.

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 Verlag, 1997), pp. 81–104.

Ojeda-Castañeda, J.

Pe’er, A.

D. Y. 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]

Primot, J.

N. Guérineau, B. Harchaoui, J. Primot, K. Heggarty, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. Lett. 26, 411–423 (2001).
[CrossRef]

N. Guérineau, B. Harchaoui, J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199–203 (2000).
[CrossRef]

Quintero, O.

A. Kolodziejczyk, Z. Jaroszewicz, R. Henao, O. Quintero, “The Talbot array illuminator: imaging properties and a new interpretation,” J. Opt. A Pure Appl. Opt. 6, 651–657 (2004).
[CrossRef]

A. Kolodziejczyk, Z. Jaroszewicz, A. Kowalik, O. Quintero, “Kinoform sampling filter,” Opt. Commun. 200, 35–42 (2001).
[CrossRef]

Saavedra, G.

Sicre, E. E.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Sinzinger, S.

S. Sinzinger, J. Jahns, Microoptics (Wiley-VCH, 2003).
[CrossRef]

Swanson, G. J.

Tajahuerce, E.

J. Lancis, G. Mínguez-Vega, E. Tajahuerce, V. Climent, P. Andrés, J. Caraquitena, “Chromatic compensation of broadband light diffraction: ABCD-matrix approach,” J. Opt. Soc. Am. A 21, 1875–1885 (2004).
[CrossRef]

J. Lancis, G. Mínguez-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. Mínguez-Vega, J. Lancis, E. Tajahuerce, V. Climent, J. Caraquitena, P. Andrés, “Broadband space-variant Fresnel processor,” Opt. Lett. 27, 1926–1928 (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. Mínguez-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]

E. Tajahuerce, E. Bonet, P. Andrés, C. J. Zapata-Rodríguez, 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]

Thomas, J. A.

Wang, D. Y.

D. Y. 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]

Zapata-Rodríguez, C. J.

Zweig, D. A.

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed.(Academic, 1987), Chap. 1.2.

Appl. Opt. (6)

J. Mod. Opt. (1)

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. Opt. A Pure Appl. Opt. (1)

A. Kolodziejczyk, Z. Jaroszewicz, R. Henao, O. Quintero, “The Talbot array illuminator: imaging properties and a new interpretation,” J. Opt. A Pure Appl. Opt. 6, 651–657 (2004).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (4)

A. Kolodziejczyk, Z. Jaroszewicz, A. Kowalik, O. Quintero, “Kinoform sampling filter,” Opt. Commun. 200, 35–42 (2001).
[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. Tajahuerce, P. Andrés, G. Mínguez-Vega, M. Fernández-Alonso, V. Climent, “Quasi-wavelength-independent broadband optical Fourier transformer,” Opt. Commun. 172, 153–160 (1999).
[CrossRef]

N. Guérineau, B. Harchaoui, J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199–203 (2000).
[CrossRef]

Opt. Eng. (1)

D. Y. 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]

Opt. Lett. (5)

Other (6)

A. E. Siegman, Lasers (University Science, 1986).

J. M. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).
[CrossRef]

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed.(Academic, 1987), Chap. 1.2.

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 Verlag, 1997), pp. 81–104.

S. Sinzinger, J. Jahns, Microoptics (Wiley-VCH, 2003).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Nearly chromatically compensated Fresnel diffraction setup: diffractive lens triplet. When lens DL1 is removed, the diffractive lens doublet DL2 and DL3 makes up an achromatic Fourier transformer. S, broadband point source.

Fig. 2
Fig. 2

Variation of diffraction efficiency of the eight-level kinoform sampling filter versus reconstruction wavenumber σ. The σ interval covers the entire white-light spectrum, and the design wavenumber is σd = 1.58 µm−1, which corresponds to the He–Ne laser wavelength.

Fig. 3
Fig. 3

(Color online) Black-and-white picture of the Fresnel-irradiance pattern with Q = − 1 and L = 0 generated by the kinoform sampling filter under (a) monochromatic illumination of wavelength 565 nm and (b) white-light illumination from a xenon-arc lamp.

Fig. 4
Fig. 4

(Color online) Black-and-white representation of the array of spots provided by the optical configuration in Fig. 1 under white-light illumination from a xenon-arc lamp for two axial positions of the filter plane: (a) z = 160 mm and (b) z = 200 mm.

Equations (14)

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

Z ( σ ) = Z 0 σ / σ 0 ,
U out ( x , y ; σ ) = σ exp ( i 2 π L σ ) i B exp [ i π σ D B ( x 2 + y 2 ) ] × U in ( x 0 , y 0 ; σ ) exp [ i π σ A B ( x 0 2 + y 0 2 ) ] exp [ i 2 π σ x 0 x + y 0 y B ] d x 0 d y 0 ,
[ A B C D ] = [ 1 0 C / A 1 ] [ A 0 0 1 / A ] [ 1 B / A 0 1 ] ,
L eq = B / A = K 1 σ , M eq = A = K 2 ,
[ A B C D ] = [ 0 B C D ] [ 1 0 1 / f ( σ ) 1 ] = [ B / f ( σ ) B C D / f ( σ ) D ] ,
L eq = B A = f ( σ ) = f 0 σ σ 0 , M eq = A = B f ( σ ) = B σ σ f 0 .
d = Z 0 Z 0 , d = d 2 d + 2 Z 0 .
[ A B C D ] = [ 1 d 0 1 ] [ 1 0 1 / Z ( σ ) 1 ] [ 1 d 0 1 ] × [ 1 0 1 / Z ( σ ) 1 ] [ 1 z d 0 1 ] × [ 1 0 1 / f ( σ ) 1 ] [ 1 0 1 / z 1 ] .
σ ( L eq σ ) | σ 0 = 0 , σ M eq | σ 0 = 0 .
L eq σ = f 0 σ 0 ,
M eq = Z 0 z f 0 ( 2 Z 0 + d ) .
R d σ d = 2 [ ( Q + I J ) 1 2 N ] p 2 ,
η ( σ ) = 100 % N 4 | H = 0 N 1 K = 0 N 1 exp [ i π N ( H 2 + K 2 ) ] × exp [ i 2 π L σ σ d m ( H , K ) ] | 2 ,
π N ( H 2 + K 2 ) | mod 2 π × [ 2 π m ( H , K ) L , 2 π [ m ( H , K ) + 1 ] L ] .

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