Abstract

We present a general procedure to analyze the axial-irradiance distribution generated by an unlimited diffractive lens under coherent, Gaussian illumination. The resulting on-axis diffraction pattern, which is evaluated in terms of the power complex spectrum of the Fresnel-zone transmittance, explicitly depends on the truncation parameter that we define, which evaluates the effective number of zones illuminated by the Gaussian beam. Depending on the value of this parameter, different kinds of axial behavior are observed. In particular, for moderate values a multiple-focal-shift phenomenon appears, and a simple formula for its evaluation is presented. Additionally, for low values of the truncation parameter, a focal-merge effect is observed in multifocal zone plates.

© 1999 Optical Society of America

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References

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  1. Feature issue on diffractive optics applications, Appl. Opt. 34, 2399–2559 (1995).
  2. Y. Ono, N. Nishida, “Holographic laser scanners using generalized zone plates,” Appl. Opt. 21, 4542–4548 (1982).
    [CrossRef] [PubMed]
  3. S. L. Dobson, P.-C. Sun, Y. Fainman, “Diffractive lenses for chromatic confocal imaging,” Appl. Opt. 36, 4744–4748 (1997).
    [CrossRef] [PubMed]
  4. P. C. Lin, P.-C. Sun, L. Zhu, Y. Fainman, “Single-shot depth-section imaging through chromatic slit-scan confocal microscopy,” Appl. Opt. 37, 6764–6770 (1998).
    [CrossRef]
  5. M. Rossi, T. Hesser, “Interference effects in diffractive beam shaping elements,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper DMD3-1.
  6. D. G. Greywall, “A micromechanical optical switch with a zone-plate reflector,” J. Phys. D 30, 2191–2208 (1987).
    [CrossRef]
  7. V. P. Koronkevich, V. P. Kiriyanov, F. I. Kokoulin, I. G. Palchikova, A. G. Poleshchuk, A. G. Sedukhin, E. G. Churin, A. M. Shcherbachenko, Y. I. Yurlov, “Fabrication of kinoform optical elements,” Optik (Stuttgart) 67, 257–266 (1984).
  8. V. Moreno, J. F. Román, J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
    [CrossRef]
  9. D. L. Dichensheets, G. S. Kino, “Micromachined scanning confocal microscope,” Opt. Lett. 21, 764–766 (1996).
    [CrossRef]
  10. A. Boivin, “On the theory of diffraction by concentric arrays of ring-shaped apertures,” J. Opt. Soc. Am. 42, 60–64 (1952).
    [CrossRef]
  11. M. Novotný, “A new series representation of the Fresnel diffraction field of axially symmetrical filters,” Opt. Acta 24, 551–565 (1977).
    [CrossRef]
  12. R. Chmelik, “Analytic description of wave fields in focal regions of diffractive lenses,” J. Mod. Opt. 43, 1463–1471 (1996).
    [CrossRef]
  13. V. P. Koronkevich, I. G. Pal’chikova, “Modern zone plates,” Optoelectron. Instrum. Data Process. 1, 86–100 (1992).
  14. X. Jiang, S. Wang, E. Bernabeu, J. Alda, “ABCD matrix and focal shift for Fresnel zone plates,” Optik (Stuttgart) 95, 16–18 (1993).
  15. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  16. M. Martı́nez-Corral, C. J. Zapata-Rodrı́guez, P. Andrés, E. Silvestre, “Effective Fresnel-number concept for evaluating the relative focal shift in focused beams,” J. Opt. Soc. Am. A 15, 449–455 (1998).
    [CrossRef]
  17. S. De Nicola, D. Anderson, M. Lisak, “Focal shift effects in diffracted focused beams,” Pure Appl. Opt. 7, 1249–1259 (1998).
    [CrossRef]
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1996), Sec. 3.4.2.
  19. C. W. McCutchen, “Generalized aperture and three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  20. M. Martı́nez-Corral, P. Andrés, J. Ojeda-Castañeda, “On-axis diffractional behavior of two-dimensional pupils,” Appl. Opt. 33, 2223–2229 (1994).
    [CrossRef] [PubMed]
  21. S. Wang, E. Bernabeu, J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351–1358 (1992).
    [CrossRef]
  22. Hint: |sin(x+iy)|2=sin2(x)+sinh2(y).
  23. P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 11.
  24. A. R. Shulman, Optical Data Processing (Wiley, New York, 1970), Chap. 9.
  25. W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989–1994 (1982).
    [CrossRef] [PubMed]
  26. D. W. Sweeney, G. E. Sommargren, “Harmonic diffractive lenses,” Appl. Opt. 34, 2469–2475 (1995).
    [CrossRef] [PubMed]
  27. Y. Li, “Focal shift and focal switch in dual-focus systems,” J. Opt. Soc. Am. A 14, 1297–1304 (1997).
    [CrossRef]
  28. M. Martı́nez-Corral, V. Climent, “Focal switch: a new effect in low-Fresnel-number systems,” Appl. Opt. 35, 24–27 (1996).
    [CrossRef] [PubMed]

1998 (3)

1997 (3)

1996 (3)

1995 (2)

1994 (1)

1993 (1)

X. Jiang, S. Wang, E. Bernabeu, J. Alda, “ABCD matrix and focal shift for Fresnel zone plates,” Optik (Stuttgart) 95, 16–18 (1993).

1992 (2)

V. P. Koronkevich, I. G. Pal’chikova, “Modern zone plates,” Optoelectron. Instrum. Data Process. 1, 86–100 (1992).

S. Wang, E. Bernabeu, J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351–1358 (1992).
[CrossRef]

1987 (1)

D. G. Greywall, “A micromechanical optical switch with a zone-plate reflector,” J. Phys. D 30, 2191–2208 (1987).
[CrossRef]

1984 (1)

V. P. Koronkevich, V. P. Kiriyanov, F. I. Kokoulin, I. G. Palchikova, A. G. Poleshchuk, A. G. Sedukhin, E. G. Churin, A. M. Shcherbachenko, Y. I. Yurlov, “Fabrication of kinoform optical elements,” Optik (Stuttgart) 67, 257–266 (1984).

1982 (2)

1981 (1)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

1977 (1)

M. Novotný, “A new series representation of the Fresnel diffraction field of axially symmetrical filters,” Opt. Acta 24, 551–565 (1977).
[CrossRef]

1964 (1)

1952 (1)

Alda, J.

X. Jiang, S. Wang, E. Bernabeu, J. Alda, “ABCD matrix and focal shift for Fresnel zone plates,” Optik (Stuttgart) 95, 16–18 (1993).

S. Wang, E. Bernabeu, J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351–1358 (1992).
[CrossRef]

Anderson, D.

S. De Nicola, D. Anderson, M. Lisak, “Focal shift effects in diffracted focused beams,” Pure Appl. Opt. 7, 1249–1259 (1998).
[CrossRef]

Andrés, P.

Bernabeu, E.

X. Jiang, S. Wang, E. Bernabeu, J. Alda, “ABCD matrix and focal shift for Fresnel zone plates,” Optik (Stuttgart) 95, 16–18 (1993).

S. Wang, E. Bernabeu, J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351–1358 (1992).
[CrossRef]

Boivin, A.

Carter, W. H.

Chmelik, R.

R. Chmelik, “Analytic description of wave fields in focal regions of diffractive lenses,” J. Mod. Opt. 43, 1463–1471 (1996).
[CrossRef]

Churin, E. G.

V. P. Koronkevich, V. P. Kiriyanov, F. I. Kokoulin, I. G. Palchikova, A. G. Poleshchuk, A. G. Sedukhin, E. G. Churin, A. M. Shcherbachenko, Y. I. Yurlov, “Fabrication of kinoform optical elements,” Optik (Stuttgart) 67, 257–266 (1984).

Climent, V.

De Nicola, S.

S. De Nicola, D. Anderson, M. Lisak, “Focal shift effects in diffracted focused beams,” Pure Appl. Opt. 7, 1249–1259 (1998).
[CrossRef]

Dichensheets, D. L.

Dobson, S. L.

Eberly, J. H.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 11.

Fainman, Y.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1996), Sec. 3.4.2.

Greywall, D. G.

D. G. Greywall, “A micromechanical optical switch with a zone-plate reflector,” J. Phys. D 30, 2191–2208 (1987).
[CrossRef]

Hesser, T.

M. Rossi, T. Hesser, “Interference effects in diffractive beam shaping elements,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper DMD3-1.

Jiang, X.

X. Jiang, S. Wang, E. Bernabeu, J. Alda, “ABCD matrix and focal shift for Fresnel zone plates,” Optik (Stuttgart) 95, 16–18 (1993).

Kino, G. S.

Kiriyanov, V. P.

V. P. Koronkevich, V. P. Kiriyanov, F. I. Kokoulin, I. G. Palchikova, A. G. Poleshchuk, A. G. Sedukhin, E. G. Churin, A. M. Shcherbachenko, Y. I. Yurlov, “Fabrication of kinoform optical elements,” Optik (Stuttgart) 67, 257–266 (1984).

Kokoulin, F. I.

V. P. Koronkevich, V. P. Kiriyanov, F. I. Kokoulin, I. G. Palchikova, A. G. Poleshchuk, A. G. Sedukhin, E. G. Churin, A. M. Shcherbachenko, Y. I. Yurlov, “Fabrication of kinoform optical elements,” Optik (Stuttgart) 67, 257–266 (1984).

Koronkevich, V. P.

V. P. Koronkevich, I. G. Pal’chikova, “Modern zone plates,” Optoelectron. Instrum. Data Process. 1, 86–100 (1992).

V. P. Koronkevich, V. P. Kiriyanov, F. I. Kokoulin, I. G. Palchikova, A. G. Poleshchuk, A. G. Sedukhin, E. G. Churin, A. M. Shcherbachenko, Y. I. Yurlov, “Fabrication of kinoform optical elements,” Optik (Stuttgart) 67, 257–266 (1984).

Li, Y.

Y. Li, “Focal shift and focal switch in dual-focus systems,” J. Opt. Soc. Am. A 14, 1297–1304 (1997).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Lin, P. C.

Lisak, M.

S. De Nicola, D. Anderson, M. Lisak, “Focal shift effects in diffracted focused beams,” Pure Appl. Opt. 7, 1249–1259 (1998).
[CrossRef]

Marti´nez-Corral, M.

McCutchen, C. W.

Milonni, P. W.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 11.

Moreno, V.

V. Moreno, J. F. Román, J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

Nishida, N.

Novotný, M.

M. Novotný, “A new series representation of the Fresnel diffraction field of axially symmetrical filters,” Opt. Acta 24, 551–565 (1977).
[CrossRef]

Ojeda-Castañeda, J.

Ono, Y.

Pal’chikova, I. G.

V. P. Koronkevich, I. G. Pal’chikova, “Modern zone plates,” Optoelectron. Instrum. Data Process. 1, 86–100 (1992).

Palchikova, I. G.

V. P. Koronkevich, V. P. Kiriyanov, F. I. Kokoulin, I. G. Palchikova, A. G. Poleshchuk, A. G. Sedukhin, E. G. Churin, A. M. Shcherbachenko, Y. I. Yurlov, “Fabrication of kinoform optical elements,” Optik (Stuttgart) 67, 257–266 (1984).

Poleshchuk, A. G.

V. P. Koronkevich, V. P. Kiriyanov, F. I. Kokoulin, I. G. Palchikova, A. G. Poleshchuk, A. G. Sedukhin, E. G. Churin, A. M. Shcherbachenko, Y. I. Yurlov, “Fabrication of kinoform optical elements,” Optik (Stuttgart) 67, 257–266 (1984).

Román, J. F.

V. Moreno, J. F. Román, J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

Rossi, M.

M. Rossi, T. Hesser, “Interference effects in diffractive beam shaping elements,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper DMD3-1.

Salgueiro, J. R.

V. Moreno, J. F. Román, J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

Sedukhin, A. G.

V. P. Koronkevich, V. P. Kiriyanov, F. I. Kokoulin, I. G. Palchikova, A. G. Poleshchuk, A. G. Sedukhin, E. G. Churin, A. M. Shcherbachenko, Y. I. Yurlov, “Fabrication of kinoform optical elements,” Optik (Stuttgart) 67, 257–266 (1984).

Shcherbachenko, A. M.

V. P. Koronkevich, V. P. Kiriyanov, F. I. Kokoulin, I. G. Palchikova, A. G. Poleshchuk, A. G. Sedukhin, E. G. Churin, A. M. Shcherbachenko, Y. I. Yurlov, “Fabrication of kinoform optical elements,” Optik (Stuttgart) 67, 257–266 (1984).

Shulman, A. R.

A. R. Shulman, Optical Data Processing (Wiley, New York, 1970), Chap. 9.

Silvestre, E.

Sommargren, G. E.

Sun, P.-C.

Sweeney, D. W.

Wang, S.

X. Jiang, S. Wang, E. Bernabeu, J. Alda, “ABCD matrix and focal shift for Fresnel zone plates,” Optik (Stuttgart) 95, 16–18 (1993).

S. Wang, E. Bernabeu, J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351–1358 (1992).
[CrossRef]

Wolf, E.

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Yurlov, Y. I.

V. P. Koronkevich, V. P. Kiriyanov, F. I. Kokoulin, I. G. Palchikova, A. G. Poleshchuk, A. G. Sedukhin, E. G. Churin, A. M. Shcherbachenko, Y. I. Yurlov, “Fabrication of kinoform optical elements,” Optik (Stuttgart) 67, 257–266 (1984).

Zapata-Rodri´guez, C. J.

Zhu, L.

Am. J. Phys. (1)

V. Moreno, J. F. Román, J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

Appl. Opt. (8)

J. Mod. Opt. (1)

R. Chmelik, “Analytic description of wave fields in focal regions of diffractive lenses,” J. Mod. Opt. 43, 1463–1471 (1996).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

D. G. Greywall, “A micromechanical optical switch with a zone-plate reflector,” J. Phys. D 30, 2191–2208 (1987).
[CrossRef]

Opt. Acta (1)

M. Novotný, “A new series representation of the Fresnel diffraction field of axially symmetrical filters,” Opt. Acta 24, 551–565 (1977).
[CrossRef]

Opt. Commun. (1)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

S. Wang, E. Bernabeu, J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351–1358 (1992).
[CrossRef]

Optik (Stuttgart) (2)

V. P. Koronkevich, V. P. Kiriyanov, F. I. Kokoulin, I. G. Palchikova, A. G. Poleshchuk, A. G. Sedukhin, E. G. Churin, A. M. Shcherbachenko, Y. I. Yurlov, “Fabrication of kinoform optical elements,” Optik (Stuttgart) 67, 257–266 (1984).

X. Jiang, S. Wang, E. Bernabeu, J. Alda, “ABCD matrix and focal shift for Fresnel zone plates,” Optik (Stuttgart) 95, 16–18 (1993).

Optoelectron. Instrum. Data Process. (1)

V. P. Koronkevich, I. G. Pal’chikova, “Modern zone plates,” Optoelectron. Instrum. Data Process. 1, 86–100 (1992).

Pure Appl. Opt. (1)

S. De Nicola, D. Anderson, M. Lisak, “Focal shift effects in diffracted focused beams,” Pure Appl. Opt. 7, 1249–1259 (1998).
[CrossRef]

Other (5)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1996), Sec. 3.4.2.

M. Rossi, T. Hesser, “Interference effects in diffractive beam shaping elements,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper DMD3-1.

Hint: |sin(x+iy)|2=sin2(x)+sinh2(y).

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 11.

A. R. Shulman, Optical Data Processing (Wiley, New York, 1970), Chap. 9.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the optical arrangement.

Fig. 2
Fig. 2

Normalized interference term versus reduced axial coordinate N/2πα for different values of the truncation parameter α. The peak structures of this term quickly disappear for decreasing values of the truncation parameter close to unity.

Fig. 3
Fig. 3

Axial-irradiance distribution, normalized to the in-focus value, corresponding to a converging kinoform zone plate (m=1) illuminated by a Gaussian beam. We observe a focal-shift effect when the truncation parameter decreases to values close to unity. In this case the point of maximum irradiance along the axis has a Gaussian Fresnel number higher than that corresponding to the principal focus, meaning that the focus shifts toward the kinoform lens.

Fig. 4
Fig. 4

Normalized axial-irradiance distribution versus reduced Gaussian Fresnel number. We observe a deviation from the multifocal behavior for truncation parameters close to unity. The height of the peaks does not equalize, since the propagation factor causes an asymmetry along the optic axis. Also, this term leads to the displacement of the axial maxima toward the Fresnel zone plate, giving rise to a multiple focal shift.

Fig. 5
Fig. 5

Graphical representation of the axial maxima positions for the first three diffracting orders in terms of the truncation parameter. We represent the approximated expression given in Eq. (16) by dashed curves.

Fig. 6
Fig. 6

Axial-irradiance distribution versus normalized Fresnel number of a Fresnel zone plate illuminated with a plane Gaussian beam for values of α=10, where we clearly observe focal-peak generation, α=1/8, where the first- and third-order maxima merge, and α=0, corresponding to an ideally nontruncated Gaussian beam. For convenience, the normalization of the axial-irradiance distribution is different for the cases presented here.

Fig. 7
Fig. 7

Normalized axial-irradiance distribution for a kinoform zone plate with a stepwise profile. We have selected (a) M=2 and (b) M=4 phase steps as examples. We observe that for increasing number of steps, successive diffracting orders vanish. Thus, for the limiting case of an infinite number of steps, we generate a single-focus diffractive lens.

Fig. 8
Fig. 8

Position of the maxima corresponding to the axial-irradiance distribution in terms of the truncation parameter. The geometrical parameters of the stepwise relief diffractive lens coincide with those of Fig. 7.

Equations (31)

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t(ρ)=n=0aρ2T2-n=aρ2T2n=0δ(ρ2-nT2),
U0(ρ)=A exp-ρ2ω2t(ρ),
U(r, z)=exp(ikz)iλz expi k2z r20U0(ρ)×expi k2z ρ2J02π rρλz2πρ dρ.
U(z)=A πλz -q(ζ)expi2π12λz+i 12πω2ζdζ,
q(ζ)=aζT2n=0δ(ζ-nT2).
0n=0δ(ζ-nT2)expi2π12λz+i 12πω2ζdζ
=n=0 expi2πnT22λz+i T22πω2.
n=0xn=11-x,
F(N, α)11-expi2πN+i2πα,
N=π ω2λz,
α=ω2T2.
I(N)=Nα2a˜N+i2πα2|F(N, α)|2,
|F(N, α)|2=14 exp(1/α)sin2(N/2α)+sinh2(1/2α).
Nn=2πnαwithn=0, ±1, ±2, .
F=π2 1sinh(1/2α).
Fn(N, α)=i2π[(N-Nn)+i]/(2πα).
In(N)=|a˜(n)|2 N2(N-Nn)2+1.
Nmax(n)=Nn1+1Nn2=Nn1+1(2πnα)2.
a(ξ)=exp(-i2πmξ),
sinc(x)=sin(πx)πx,
aN+i2πα2=sincN+i2πα-m2=(2α)2exp(1/α) sin2(N/2α)+sinh2(1/2α)(N-Nm)2+1.
I(N)=N2(N-Nm)2+1.
a(ξ)=10ξ<1/20otherwise.
a˜(u)=12 expi π2 usincu2.
I(N)=exp(1/2α) N21+N2 sin2(N/4α)+sinh2(1/4α)sin2(N/2α)+sinh2(1/2α).
a(ξ)=exp[-i2πϕ(ξ)],
ϕ(ξ)=00ξ<1/M1/M1/Mξ<2/M2/M2/Mξ<3/M(M-1)/M,(M-1)/Mξ1.
a˜(u)=01a(ζ)exp(i2πuζ)dζ=n=0M-1 exp-i2π nMn/M(n+1)/M exp(i2πuζn)dζn.
a˜(u)=n=0M-1 exp-i2π nMexpi2π nM u×01/M exp(i2πuζ)dζ.
a˜(u)=expi πMexp[iπ(u-1)]×sinc(u-1)sinc[(u-1)/M] sinc(u/M).
I(N)=N21+N2 sin2(N/2Mα)+sinh2(1/2Mα)sin2(N/2Mα-π/M)+sinh2(1/2Mα).

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