Abstract

This study introduces and examines the diffraction properties of a so-called petal-like zone plate, which comprises Fresnel zones analogous to petals. We show that the focusing behavior of this novel type of zone plate depends on the number of petals included in the element. For a small value of the petal frequency, we observe star-like diffraction patterns at the focal plane, whose number of star arms equals the petal frequency of the element when the frequency is an odd integer and is twice as large as the petal frequency when it is an even number. In addition, we have shown that the star-like pattern rotates when it passes through the focus. Moreover, it is demonstrated that the element acts as a long depth bifocal diffractive lens for a large value of the petal frequency. The spacing between the foci is simply controlled by a so-called shifting parameter. At the same time, an annular beam is observed in the middle of the line joining the two foci together. Consequently, an axial bottle-like beam is produced around the focus, whose size could be simply monitored. Simulation results are followed and verified by experimental works.

© 2018 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]
  8. A. Davison and M. J. Simpson, “History and development of the apodized diffractive intraocular lens,” J. Cataract. Refract. Surg. 32, 849–858 (2006).
    [Crossref]
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    [Crossref]
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2017 (3)

2015 (2)

2014 (1)

2012 (1)

2009 (3)

J. Alda, J. M. Rico-Garca, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
[Crossref]

J. Alda and F. J. Gonzalez, “Polygonal Fresnel zone plates,” J. Opt. A 11, 085707 (2009).
[Crossref]

W. Chao, J. Kim, S. Rekawa, P. Fischer, and E. H. Anderson, “Demonstration of 12  nm resolution Fresnel zone plate lens based soft X-ray microscopy,” Opt. Express 17, 17669–17677 (2009).
[Crossref]

2006 (1)

A. Davison and M. J. Simpson, “History and development of the apodized diffractive intraocular lens,” J. Cataract. Refract. Surg. 32, 849–858 (2006).
[Crossref]

2004 (1)

2003 (1)

2002 (2)

C. Pati, G. S. Chen, C. G. Konkola, P. T. Heilmann, R. K. Schattenburg, M. L. Liddle, and A. Anderson, “Precision fringe metrology using a Fresnel zone plate,” J. Vac. Sci. Technol. B. 20, 2617–3079 (2002).
[Crossref]

S. Wang and X. Zhang, “Terahertz technology: terahertz tomographic imaging with a Fresnel lens,” Opt. Photonics. News 13(12), 58 (2002).
[Crossref]

1992 (2)

1987 (2)

W. B. Yun and M. R. Howells, “High-resolution Fresnel zone plates for x-ray applications by spatial-frequency multiplication,” J. Opt. Soc. Am. A 4, 34–40 (1987).
[Crossref]

D. Black and J. Wiltse, “Millimeter-wave characteristics of phase-correcting Fresnel zone plates,” IEEE Trans. Microw. Theory Tech. 35, 1122–1129 (1987).
[Crossref]

1980 (1)

1973 (1)

K. Dey and P. Khastgir, “A study of the characteristics of a microwave spherical zone plate antenna,” Int. J. Electron. 35, 97–103 (1973).
[Crossref]

Alda, J.

J. Alda, J. M. Rico-Garca, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
[Crossref]

J. Alda and F. J. Gonzalez, “Polygonal Fresnel zone plates,” J. Opt. A 11, 085707 (2009).
[Crossref]

F. J. Gonzalez, J. Alda, B. Ilic, and G. Boreman, “Infrared antennas coupled to lithographic Fresnel zone plate lenses,” Appl. Opt. 43, 6067–6073 (2004).
[Crossref]

Anderson, A.

C. Pati, G. S. Chen, C. G. Konkola, P. T. Heilmann, R. K. Schattenburg, M. L. Liddle, and A. Anderson, “Precision fringe metrology using a Fresnel zone plate,” J. Vac. Sci. Technol. B. 20, 2617–3079 (2002).
[Crossref]

Anderson, E. H.

Arfken, G. B.

H. J. Weber and G. B. Arfken, Essential Mathematical Methods for Physicists, 6th ed. (Academic, 2003).

Beynon, T. D.

Black, D.

D. Black and J. Wiltse, “Millimeter-wave characteristics of phase-correcting Fresnel zone plates,” IEEE Trans. Microw. Theory Tech. 35, 1122–1129 (1987).
[Crossref]

Boreman, G.

Chao, W.

Chen, G. S.

C. Pati, G. S. Chen, C. G. Konkola, P. T. Heilmann, R. K. Schattenburg, M. L. Liddle, and A. Anderson, “Precision fringe metrology using a Fresnel zone plate,” J. Vac. Sci. Technol. B. 20, 2617–3079 (2002).
[Crossref]

Cuadrado, J. M.

Davison, A.

A. Davison and M. J. Simpson, “History and development of the apodized diffractive intraocular lens,” J. Cataract. Refract. Surg. 32, 849–858 (2006).
[Crossref]

Dey, K.

K. Dey and P. Khastgir, “A study of the characteristics of a microwave spherical zone plate antenna,” Int. J. Electron. 35, 97–103 (1973).
[Crossref]

Du, Y.

Fischer, P.

Furlan, W. D.

Gao, N.

Gomez-Reino, C.

Gonzalez, F. J.

Heckenberg, N. R.

Heilmann, P. T.

C. Pati, G. S. Chen, C. G. Konkola, P. T. Heilmann, R. K. Schattenburg, M. L. Liddle, and A. Anderson, “Precision fringe metrology using a Fresnel zone plate,” J. Vac. Sci. Technol. B. 20, 2617–3079 (2002).
[Crossref]

Howells, M. R.

Hua, Y.

Ilic, B.

Jia, W.

Khastgir, P.

K. Dey and P. Khastgir, “A study of the characteristics of a microwave spherical zone plate antenna,” Int. J. Electron. 35, 97–103 (1973).
[Crossref]

Kim, J.

Kirk, I.

Konkola, C. G.

C. Pati, G. S. Chen, C. G. Konkola, P. T. Heilmann, R. K. Schattenburg, M. L. Liddle, and A. Anderson, “Precision fringe metrology using a Fresnel zone plate,” J. Vac. Sci. Technol. B. 20, 2617–3079 (2002).
[Crossref]

Li, H.

Li, S.

Liang, Y.

Liddle, M. L.

C. Pati, G. S. Chen, C. G. Konkola, P. T. Heilmann, R. K. Schattenburg, M. L. Liddle, and A. Anderson, “Precision fringe metrology using a Fresnel zone plate,” J. Vac. Sci. Technol. B. 20, 2617–3079 (2002).
[Crossref]

Lu, Y.

Mathews, T. R.

McDuff, R.

Meshginqalam, B.

Monsoriu, J. A.

Pati, C.

C. Pati, G. S. Chen, C. G. Konkola, P. T. Heilmann, R. K. Schattenburg, M. L. Liddle, and A. Anderson, “Precision fringe metrology using a Fresnel zone plate,” J. Vac. Sci. Technol. B. 20, 2617–3079 (2002).
[Crossref]

Perez, M. V.

Rafighdoost, J.

Rekawa, S.

Rico-Garca, J. M.

J. Alda, J. M. Rico-Garca, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
[Crossref]

Saavedra, G.

Sabatyan, A.

Salgado-Remacha, F. J.

J. Alda, J. M. Rico-Garca, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
[Crossref]

Sanchez-Brea, L. M.

J. Alda, J. M. Rico-Garca, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
[Crossref]

Schattenburg, R. K.

C. Pati, G. S. Chen, C. G. Konkola, P. T. Heilmann, R. K. Schattenburg, M. L. Liddle, and A. Anderson, “Precision fringe metrology using a Fresnel zone plate,” J. Vac. Sci. Technol. B. 20, 2617–3079 (2002).
[Crossref]

Simpson, M. J.

A. Davison and M. J. Simpson, “History and development of the apodized diffractive intraocular lens,” J. Cataract. Refract. Surg. 32, 849–858 (2006).
[Crossref]

Smith, C. P.

Wang, E.

Wang, S.

S. Wang and X. Zhang, “Terahertz technology: terahertz tomographic imaging with a Fresnel lens,” Opt. Photonics. News 13(12), 58 (2002).
[Crossref]

Wang, Z.

Weber, H. J.

H. J. Weber and G. B. Arfken, Essential Mathematical Methods for Physicists, 6th ed. (Academic, 2003).

White, A. G.

Wiltse, J.

D. Black and J. Wiltse, “Millimeter-wave characteristics of phase-correcting Fresnel zone plates,” IEEE Trans. Microw. Theory Tech. 35, 1122–1129 (1987).
[Crossref]

Wu, J.

Xiang, C.

Xie, C.

Ye, T.

Yu, J.

Yun, W. B.

Zhang, X.

S. Wang and X. Zhang, “Terahertz technology: terahertz tomographic imaging with a Fresnel lens,” Opt. Photonics. News 13(12), 58 (2002).
[Crossref]

Zhou, C.

Zhu, L.

Appl. Opt. (4)

Chin. Opt. Lett. (1)

IEEE Trans. Microw. Theory Tech. (1)

D. Black and J. Wiltse, “Millimeter-wave characteristics of phase-correcting Fresnel zone plates,” IEEE Trans. Microw. Theory Tech. 35, 1122–1129 (1987).
[Crossref]

Int. J. Electron. (1)

K. Dey and P. Khastgir, “A study of the characteristics of a microwave spherical zone plate antenna,” Int. J. Electron. 35, 97–103 (1973).
[Crossref]

J. Cataract. Refract. Surg. (1)

A. Davison and M. J. Simpson, “History and development of the apodized diffractive intraocular lens,” J. Cataract. Refract. Surg. 32, 849–858 (2006).
[Crossref]

J. Opt. A (1)

J. Alda and F. J. Gonzalez, “Polygonal Fresnel zone plates,” J. Opt. A 11, 085707 (2009).
[Crossref]

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

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

J. Vac. Sci. Technol. B. (1)

C. Pati, G. S. Chen, C. G. Konkola, P. T. Heilmann, R. K. Schattenburg, M. L. Liddle, and A. Anderson, “Precision fringe metrology using a Fresnel zone plate,” J. Vac. Sci. Technol. B. 20, 2617–3079 (2002).
[Crossref]

Opt. Commun. (1)

J. Alda, J. M. Rico-Garca, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Opt. Photonics. News (1)

S. Wang and X. Zhang, “Terahertz technology: terahertz tomographic imaging with a Fresnel lens,” Opt. Photonics. News 13(12), 58 (2002).
[Crossref]

Optik (1)

A. Sabatyan and J. Rafighdoost, “Focusing specification of cross-like Fresnel zone plate,” Optik 126, 4796–4799 (2015).
[Crossref]

Other (1)

H. J. Weber and G. B. Arfken, Essential Mathematical Methods for Physicists, 6th ed. (Academic, 2003).

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Figures (15)

Fig. 1.
Fig. 1. Typical samples of PZPs with different petal frequencies, N=4,5, and 20 petals, are presented in the first through the third columns, and a variety of αRs, αR=0.05,0.1, and 0.2, are shown in the first to the third rows, respectively.
Fig. 2.
Fig. 2. Left: phase map of a traditional zone plate. Right: phase map of a typical PZP with N=4.
Fig. 3.
Fig. 3. Focused intensity distributions for samples with odd N=3,5,7 from left to right and different αR=0.05,0.1,0.2 from top to bottom, respectively.
Fig. 4.
Fig. 4. Focused intensity distributions for samples with even N=4,6,8 from left to right and different αR=0.05,0.1,0.2 from top to bottom, respectively.
Fig. 5.
Fig. 5. Three-dimensional (3D) representation of propagation of diffracted plane beam around the main focus. Left to right: αR=0.05,0.1,0.2, respectively. Top to bottom: N=4,9,20, respectively.
Fig. 6.
Fig. 6. Axial intensity distribution for typical PZPs having focal length f=500  mm, R=5  mm, αR=.1, and different p=20 and 50, as shown in the first and second rows, respectively.
Fig. 7.
Fig. 7. Axial intensity profile for a typical conventional zone plate and radial phase shifted zone plate having focal length F=500  mm and R=5  mm. Left figure: the conventional zone plate. Middle figure: the positive phase shifted zone plate. Right figure: the negative phase shifted zone plate.
Fig. 8.
Fig. 8. Axial intensity profile for a typical PZP having the same N=20 and different α=0.05 and 0.1 for the left and right columns, respectively.
Fig. 9.
Fig. 9. Spacing between foci (S) versus α for a variety of focal lengths: (a) f=300  (mm), (b) f=400  (mm), (c) f=500  (mm), (d) f=600  (mm), (e) f=700  (mm), and (f) f=800  (mm).
Fig. 10.
Fig. 10. Variation of the slope of the plots (M) given in Fig. 9 versus focal length (solid line) and the fitted quadratic curve (dashed line).
Fig. 11.
Fig. 11. Typical setup used in the experimental verification.
Fig. 12.
Fig. 12. Experimental intensity distributions for samples with odd N=3,5,7 from left to right and different α=0.05,0.1,0.2 from top to bottom, respectively.
Fig. 13.
Fig. 13. Experimental focused intensity distributions for samples with even N=4,6,8 from left to right and different α=0.05,0.1,0.2 from top to bottom, respectively.
Fig. 14.
Fig. 14. Simulation (the first row) and experimental (the second row) examination of the rotation of the diffraction pattern of a PZP with N=4 and α=0.1 around the focal plane.
Fig. 15.
Fig. 15. Simulation (the first row) and experimental (the second row) examination of the bifocal property of PZP having N=20 and α=0.1.

Equations (12)

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T(r,θ)=exp{iπ[rαRcos(Nθ)]2λf},
BTPZP(r,θ)=Step[T(r,θ)],
Step(x)={1imag(x)>00imag(x)<0.
U(x,y;z=f)=eikfiλfT(x,y)eik2f[(xx)2+(yy)2]dxdy,
U(ρ,ϕ,z)=T(r,θ)exp(ik2f[r22rρcos(θϕ)])rdrdθ.
exp(ixcosθ)=J0(x)+2n=1inJn(x)cos(nθ),
T(r,θ)=4eiπλf(r2+α2R2/2)m,n=0inmJm(πα2R22λf)Jn(2παRrλf)×cos(2mNθ)cos(nNθ),h(r,θ)=2eiπr2λfp=0ipJp(2πrρλf)cos[p(θϕ)],
o2πcos(qθ)cos[p(θϕ)]dθ=πcos(pϕ)δq,p,
U(ρ,ϕ)=m,n=0(1)mNimJm(πα2R22λf){i(1+N)ncos[(2mn)Nϕ]Jn(2παRrλf)J(2mn)N(2πρrλf)rdr+i(1N)ncos[(2m+n)Nϕ]Jn(2παRrλf)J(2m+n)N(2πρrλf)rdr},
Jn(x)=s=0(1)s(x/2)n+2ss!(n+s)!,
U(ρ,ϕ)=m=0Cm{2cos(2mN)J2mN(2πρrλf)+i[iNcos[(2m1)Nϕ](παRrλf)J(2m1)N(2πρrλf)rdr+iNcos[(2m+1)Nϕ](παRrλf)J(2m+1)N(2πρrλf)rdr]+O[(παRrλf)2]},
I(ρ,ϕ,z)|U(ρ,ϕ,z)|2,