Abstract

A series-form expression for the individual diffracted field of a general annular ring is derived from the Rayleigh–Sommerfeld diffraction integral. It can be used for the accurate and fast simulation of any diffractive focusing element composed of concentric transparent rings. We present a comprehensive analysis, based on the leading term and the linear superposition principle, of the focusing performances of various Fresnel zone plates. Many problems, such as the equivalent aperture function, the diffraction efficiency, the focal spot pattern, the suppression of higher orders and the appearance of “fractional orders,” and the explanation for the appearance of Fraunhofer diffraction patterns, are analytically investigated in detail. Because of the great similarity between Fresnel zone plates and multilevel diffractive lenses, most of the obtained results are also applicable to multilevel diffractive lenses.

© 2004 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2003

2002

2001

U. Levy, D. Mendlovic, E. Marom, “Efficiency analysis of diffractive lenses,” J. Opt. Soc. Am. A 18, 86–93 (2001).
[CrossRef]

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature (London) 414, 184–188 (2001).
[CrossRef]

2000

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

1999

1998

1996

C. Vassallo, “Limitations of the wide-angle beam propagation method in nonuniform systems,” J. Opt. Soc. Am. A 13, 761–770 (1996).
[CrossRef]

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

1995

E. H. Anderson, V. Boegli, L. P. Muray, “Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[CrossRef]

C. Vassallo, “Wide-angle BPM and power conservation,” IEE Electron. Lett. 31, 130–131 (1995).
[CrossRef]

M. Kuittinen, H. P. Herzig, “Encoding of efficient diffractive microlenses,” Opt. Lett. 20, 2156–2158 (1995).
[CrossRef] [PubMed]

1994

1992

1991

1990

1984

M. J. Simpson, A. G. Michette, “Imaging properties of modified Fresnel zone plates,” Opt. Acta 31, 403–413 (1984).
[CrossRef]

1983

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1981

1979

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

1977

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

1975

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1973

1972

1969

1968

1967

1966

1961

1952

Adelung, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Anderson, E. H.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

E. H. Anderson, V. Boegli, L. P. Muray, “Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[CrossRef]

Arsenault, H.

Artzner, G. E.

G. E. Artzner, J. P. Delaboudinière, X. Y. Song, “Photon sieves as EUV telescopes for solar orbiter,” in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. L. Keil, S. V. Avakyan, S. I. Vavilov, eds., Proc. SPIE4853, 158–161 (2003).
[CrossRef]

Ashman, R.

Attwood, D.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Baez, A. V.

Berndt, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Boegli, V.

E. H. Anderson, V. Boegli, L. P. Muray, “Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[CrossRef]

Boivin, A.

Bottema, M.

Cai, A.

Cao, Q.

Chao, W.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Chmeli´k, R.

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

Christ, O.

G. Schmahl, D. Rudolph, P. Guttmann, O. Christ, “Zone plates for x-ray microscopy,” in X-ray Microscopy, G. Schmahl, D. Rudolph, eds. (Springer-Verlag, Berlin, 1984), Vol. 43, pp. 63–74.

Delaboudinière, J. P.

G. E. Artzner, J. P. Delaboudinière, X. Y. Song, “Photon sieves as EUV telescopes for solar orbiter,” in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. L. Keil, S. V. Avakyan, S. I. Vavilov, eds., Proc. SPIE4853, 158–161 (2003).
[CrossRef]

Denbeaux, G.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Deng, X.

Ferris, L. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Subsect. 2.1.5.

Gu, M.

Guttmann, P.

G. Schmahl, D. Rudolph, P. Guttmann, O. Christ, “Zone plates for x-ray microscopy,” in X-ray Microscopy, G. Schmahl, D. Rudolph, eds. (Springer-Verlag, Berlin, 1984), Vol. 43, pp. 63–74.

Harm, S.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Harteneck, B.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Herzig, H. P.

Hrynevych, M.

Jahns, J.

Johnson, L.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Johnson, R. L.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Kipp, L.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Kuittinen, M.

Lax, M.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Levy, U.

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lucero, A.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Marom, E.

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mendlovic, D.

Michette, A. G.

M. J. Simpson, A. G. Michette, “Imaging properties of modified Fresnel zone plates,” Opt. Acta 31, 403–413 (1984).
[CrossRef]

Mittra, R.

Muray, L. P.

E. H. Anderson, V. Boegli, L. P. Muray, “Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[CrossRef]

Novotny, M.

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

Olynick, D. L.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Pfeifer, C. D.

Rudolph, D.

G. Schmahl, D. Rudolph, P. Guttmann, O. Christ, “Zone plates for x-ray microscopy,” in X-ray Microscopy, G. Schmahl, D. Rudolph, eds. (Springer-Verlag, Berlin, 1984), Vol. 43, pp. 63–74.

Schmahl, G.

G. Schmahl, D. Rudolph, P. Guttmann, O. Christ, “Zone plates for x-ray microscopy,” in X-ray Microscopy, G. Schmahl, D. Rudolph, eds. (Springer-Verlag, Berlin, 1984), Vol. 43, pp. 63–74.

Seemann, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Semonin, R. G.

Sheppard, C. J. R.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Simpson, M. J.

M. J. Simpson, A. G. Michette, “Imaging properties of modified Fresnel zone plates,” Opt. Acta 31, 403–413 (1984).
[CrossRef]

Sinzinger, S.

S. Sinzinger, J. Jahns, Microoptics, 2nd ed. (Wiley-VCH, Weinheim, Germany2003), Subsect. 6.3.6.

Skibowski, M.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Song, X. Y.

G. E. Artzner, J. P. Delaboudinière, X. Y. Song, “Photon sieves as EUV telescopes for solar orbiter,” in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. L. Keil, S. V. Avakyan, S. I. Vavilov, eds., Proc. SPIE4853, 158–161 (2003).
[CrossRef]

Southwell, W. H.

Stigliani, D. J.

Sun, J. A.

Vassallo, C.

Veklerov, E.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Waldman, G. S.

Walker, S. J.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, UK1966), p. 46.

Xiao, B.

Yen, W. M.

Young, M.

Am. J. Phys.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Appl. Opt.

IEE Electron. Lett.

C. Vassallo, “Wide-angle BPM and power conservation,” IEE Electron. Lett. 31, 130–131 (1995).
[CrossRef]

J. Mod. Opt.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

E. H. Anderson, V. Boegli, L. P. Muray, “Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[CrossRef]

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Nature (London)

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature (London) 414, 184–188 (2001).
[CrossRef]

Opt. Acta

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

M. J. Simpson, A. G. Michette, “Imaging properties of modified Fresnel zone plates,” Opt. Acta 31, 403–413 (1984).
[CrossRef]

Opt. Commun.

Q. Cao, X. Deng, “Power carried by scalar light beams,” Opt. Commun. 151, 212–216 (1998).
[CrossRef]

Opt. Lett.

Phys. Rev. A

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Subsect. 2.1.5.

G. E. Artzner, J. P. Delaboudinière, X. Y. Song, “Photon sieves as EUV telescopes for solar orbiter,” in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. L. Keil, S. V. Avakyan, S. I. Vavilov, eds., Proc. SPIE4853, 158–161 (2003).
[CrossRef]

M. Howells, http://www-esg.lbl.gov/esg/personnel/howells/Xraysieves.pdf . The opinion that the suppression of higher orders results from the use of different ratios d/w for different pinholes is presented in this reference, where d is the diameter of an individual pinhole and w is the width of the corresponding local half-zone of the underlying TFZP.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

S. Sinzinger, J. Jahns, Microoptics, 2nd ed. (Wiley-VCH, Weinheim, Germany2003), Subsect. 6.3.6.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, UK1966), p. 46.

G. Schmahl, D. Rudolph, P. Guttmann, O. Christ, “Zone plates for x-ray microscopy,” in X-ray Microscopy, G. Schmahl, D. Rudolph, eds. (Springer-Verlag, Berlin, 1984), Vol. 43, pp. 63–74.

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Figures (7)

Fig. 1
Fig. 1

Schematic view of a general FZP. See the text for the definitions of the parameters f, fn, and rn.

Fig. 2
Fig. 2

Phase error in the neighborhood of the main focus.

Fig. 3
Fig. 3

Normalized intensity distributions at the focal plane for three opaque-center TFZPs (to the peak intensity in each case): (a) for a TFZP with five open rings, (b) for a TFZP with ten open rings, (c) for a TFZP with 15 open rings. All the solid curves are directly calculated from Eq. (1). The dashed curves in (a), (b), and (c) are calculated by use of the first four terms, the first three terms, and the first two terms of Eq. (7) for each open ring, respectively. Note that the two curves in (a) are almost completely indistinguishable. See the text for the detailed parameters.

Fig. 4
Fig. 4

Change in the mismatched phase δn with the increase of n.

Fig. 5
Fig. 5

Normalized intensity distributions (to the peak intensity in each case) on the propagation axis: (a) for a TFZP with 300 open rings, (b) for a TFZP with 600 open rings, and (c) for a TFZP with 900 open rings. See the text for the detailed parameters.

Fig. 6
Fig. 6

Normalized intensity distributions on the propagation axis: (a) for a composite FZP, (b) for a modified composite FZP, and (c) for the MFZP presented in Ref. 22. In both (a) and (b), the intensity distributions are normalized to the main peak of (b). In (c), the intensity distribution is normalized to its own main peak. See the text for the detailed parameters.

Fig. 7
Fig. 7

(a) Change in the difference Δn with the increase of n, (b) transmittance functions at the boundaries between the inner and outer regions, (c) transmittance functions at the outermost parts of the total elements. In (b) and (c), the upper plot (with a vertical shift of 3) corresponds to the underlying transparent-center TFZP, the middle plot (with a vertical shift of 1.5) corresponds to the modified composite FZP, and the lower plot corresponds to the composite FZP. For clarity, the radial coordinates in (b) and (c) have been shifted by 23,750 nm and 71,955 nm, respectively. Note that, for the modified composite FZP, the open rings shown in (c) are the 637th, 638th, and 639th open rings of the outer region; but, for the composite FZP, those shown are the 640th, 641st, and 642nd open rings of the outer region.

Equations (40)

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Un(R)=1λAnfρ2exp(jkρ)rdrdθ,
ρfn+R2+(r2-rn2)-2Rr cos(θ-ϕ)2fn,
Un(R)=kffn2expjkfn+R22fnG(R),
G(R)=anbnexpjkr2-rn22fnJ0kRrfnrdr,
G(R)=12sn-dnsn+dnexpjks-sn2fnJ0kRfns1/2ds.
J0(s1/2)=m=01m!dmdsmJ0(s1/2)s=sn(s-sn)m,
G(R)=m=0Gm(R),
Gm(R)=jm(αR)mdnm+1m!2msnmγm+1hmJm(αR),
hm=i=0m(-1)im!(m-i)!Im[(jγ)m-i exp(jγ)],
G0(R)=dnγsin(γ)J0(αR),
G1(R)=jdn2αR2snγ2[γ cos(γ)-sin(γ)]J1(αR).
Un(R)2ffnexpjkfn-f+R22fn×sinkdn2fnJ0krnfnR,
Un(0)2ffnexp[jk(fn-f)]sinkdn2fn.
fn=f+mnλ,sinkdn2fn>0,
fn=f+mn+12λ,sinkdn2fn<0,
W(rn)=2λfFπDnfnexp[jk(fn-f)]sinkdn2fn,
U(R)=kF0AexpjkR22QW(r)J0kRQrrdr,
UL(R)=1λ0A02πQ2ρ2FW(r)exp(-jkQ)exp(jkρ)rdθdr,
U(R)2πλF0ArW(r)J02πRλFrdr,
dn=2fnkMπ-arcsinπDnfn2λfFW(rn),
W(rn)=2βλfFπD1f1exp-snm-s1mσ2m
dn=2fnkMπ-arcsinβDnfnD1f1exp-sn-s1σ2,
ηP1P2fF2A20AQ3|W(r)|2rdr.
Un(0, z)2zz2+rn2exp[jk(z2+rn2-z)]×sinkdn2z2+rn2.
δn=k3{[f2+18(n+c)λf+9(n+c)2λ2]1/2-f}-6(n+c)π,
dmdsm=ddsm=d2rdrm=12mdrdrm.
dmdsmJ0(s1/2)=2m2mdξdξmJ0(ξ).
dξdξmJ0(ξ)=(-1)mξmJm(ξ),
J0(s1/2)=m=0(-1)m(αR)mm!2msnmJm(αR)(s-sn)m,
G(R)=m=0Gm(R)=m=0jm(αR)mdnm+1m!2msnmγm+1hmJm(αR),
hm=12j-jγjγtm exp(t)dt,
hm=Im[(jγ)m exp(jγ)]-mhm-1,
Un(0, z)=1λAnzρ2exp(jkρ)rdrdθ,
ρz2+rn2+r2-rn22z2+rn2,
Un(0, z)=kzz2+rn2exp(jkz2+rn2)anbn×expjkr2-rn22z2+rn2rdr.
anbnexpjkr2-rn22z2+rn2rdr=2kz2+rn2 sinkdn2z2+rn2.
ψn(f)krn2/(2f)=2nπ+2cπ,
ψn(f)=k(f2+rn2-f)=2(n+c)π.
rn2=2(n+c)λf+(n+c)2λ2.
ψn(f/3)=k3{[f2+18(n+c)λf+9(n+c)2λ2]1/2-f}.

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