Abstract

Spoke wheel filtering strategy is developed using the proposed spoke wheel filter (SWF) and solving the nonlinear optimization model established by introducing a super-Gaussian function. Theoretical calculations through simulated annealing algorithm indicate that high focal depth is obtained by increasing the number of curved sectors composing SWF, while the peak-valley intensity oscillation is reduced by bending them appropriately and slightly changing their amplitude transmittances. The transverse in-focus spot compresses when the center-shaded circle of SWF enlarging. Comparison shows that SWF outperforms radial-symmetric pupil filters in its largely reduced intensity oscillation and very flexible design of focal depth extension.

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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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2009 (1)

2008 (2)

2007 (3)

2006 (3)

2005 (1)

R. Juškaitis, E. J. Botcherby, and T. Wilson, “Scanning microscopy with extended depth of focus,” Proc. SPIE 5701, 85–92 (2005).
[Crossref]

2001 (1)

1998 (1)

S. Yu. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7(3), 537–548 (1998).
[Crossref]

1997 (1)

R. Hild, M. J. Yzuel, and J. C. Escalera, “High focal depth imaging of small structures,” Microelectron. Eng. 34(2), 195–214 (1997).
[Crossref]

1993 (2)

J. Sochacki, Z. Jaroszewicz, L. R. Staroński, and A. Kołodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am. A 10(8), 1765–1768 (1993).
[Crossref]

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Calculation of the focusators into a longitudinal line segment and study of a focal area,” J. Mod. Opt. 40(5), 761–769 (1993).
[Crossref]

1992 (1)

H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31(Part 1, No. 12B), 4126–4130 (1992).
[Crossref]

1990 (2)

A. Kołodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, “The light sword optical element-a new diffraction structure with extended depth of focus,” J. Mod. Opt. 37(8), 1283–1286 (1990).
[Crossref]

J. Ojeda-Castaneda and L. R. Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29(7), 994–997 (1990).
[Crossref] [PubMed]

1989 (1)

1988 (1)

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

1981 (1)

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized holograms,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

1973 (1)

1960 (2)

1952 (1)

G. Toraldo di Francia, “Nuovo pupille superresolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).

Bara, S.

A. Kołodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, “The light sword optical element-a new diffraction structure with extended depth of focus,” J. Mod. Opt. 37(8), 1283–1286 (1990).
[Crossref]

Berriel-Valdos, L. R.

Botcherby, E. J.

R. Juškaitis, E. J. Botcherby, and T. Wilson, “Scanning microscopy with extended depth of focus,” Proc. SPIE 5701, 85–92 (2005).
[Crossref]

Cagigal, M. P.

Campos, J.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Two-zone pupil filters,” Opt. Commun. 281(5), 913–922 (2008).
[Crossref]

S. Ledesma, J. C. Escalera, J. Campos, J. Mazzaferri, and M. J. Yzuel, “High depth of focus by combining annular lenses,” Opt. Commun. 266(1), 6–12 (2006).
[Crossref]

Canales, V. F.

Chen, N.

Diaz, A.

Escalera, J. C.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Two-zone pupil filters,” Opt. Commun. 281(5), 913–922 (2008).
[Crossref]

S. Ledesma, J. C. Escalera, J. Campos, J. Mazzaferri, and M. J. Yzuel, “High depth of focus by combining annular lenses,” Opt. Commun. 266(1), 6–12 (2006).
[Crossref]

R. Hild, M. J. Yzuel, and J. C. Escalera, “High focal depth imaging of small structures,” Microelectron. Eng. 34(2), 195–214 (1997).
[Crossref]

Espinosa, J.

Flores, A.

Z. Liu, A. Flores, M. R. Wang, and J. J. Yang, “Diffractive infrared lens with extended depth of focus,” Opt. Eng. 46(1), 018002 (2007).
[Crossref]

Friberg, A. T.

S. Yu. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7(3), 537–548 (1998).
[Crossref]

Fukuda, H.

H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31(Part 1, No. 12B), 4126–4130 (1992).
[Crossref]

Gan, F.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

Golub, M. A.

M. A. Golub, V. Shurman, and I. Grossinger, “Extended focus diffractive optical element for Gaussian laser beams,” Appl. Opt. 45(1), 144–150 (2006).
[Crossref] [PubMed]

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized holograms,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

Grossinger, I.

Hegedus, Z. S.

Hild, R.

R. Hild, M. J. Yzuel, and J. C. Escalera, “High focal depth imaging of small structures,” Microelectron. Eng. 34(2), 195–214 (1997).
[Crossref]

Illueca, C.

Jaroszewicz, Z.

Juškaitis, R.

R. Juškaitis, E. J. Botcherby, and T. Wilson, “Scanning microscopy with extended depth of focus,” Proc. SPIE 5701, 85–92 (2005).
[Crossref]

Karpeev, S. V.

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized holograms,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Calculation of the focusators into a longitudinal line segment and study of a focal area,” J. Mod. Opt. 40(5), 761–769 (1993).
[Crossref]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

Kolodziejczyk, A.

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Calculation of the focusators into a longitudinal line segment and study of a focal area,” J. Mod. Opt. 40(5), 761–769 (1993).
[Crossref]

Ledesma, S.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Two-zone pupil filters,” Opt. Commun. 281(5), 913–922 (2008).
[Crossref]

S. Ledesma, J. C. Escalera, J. Campos, J. Mazzaferri, and M. J. Yzuel, “High depth of focus by combining annular lenses,” Opt. Commun. 266(1), 6–12 (2006).
[Crossref]

Li, G.

Lit, J. W. Y.

Liu, Z.

Z. Liu, A. Flores, M. R. Wang, and J. J. Yang, “Diffractive infrared lens with extended depth of focus,” Opt. Eng. 46(1), 018002 (2007).
[Crossref]

Mazzaferri, J.

S. Ledesma, J. C. Escalera, J. Campos, J. Mazzaferri, and M. J. Yzuel, “High depth of focus by combining annular lenses,” Opt. Commun. 266(1), 6–12 (2006).
[Crossref]

McLeod, J. H.

Mikula, G.

Moreno, I.

Ojeda-Castaneda, J.

Perez, J.

Petelczyc, K.

Popov, S. Yu.

S. Yu. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7(3), 537–548 (1998).
[Crossref]

Prokhorov, A. M.

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized holograms,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

Sheppard, C. J. R.

Shurman, V.

Singh, J.

Sisakyan, I. N.

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized holograms,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

Sochacki, J.

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Calculation of the focusators into a longitudinal line segment and study of a focal area,” J. Mod. Opt. 40(5), 761–769 (1993).
[Crossref]

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized holograms,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

Staronski, L. R.

Sypek, M.

G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, and M. Sypek, “Imaging with extended focal depth by means of lenses with radial and angular modulation,” Opt. Express 15(15), 9184–9193 (2007).
[Crossref] [PubMed]

A. Kołodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, “The light sword optical element-a new diffraction structure with extended depth of focus,” J. Mod. Opt. 37(8), 1283–1286 (1990).
[Crossref]

Tepichin, E.

Toraldo di Francia, G.

G. Toraldo di Francia, “Nuovo pupille superresolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).

Tremblay, R.

Vázquez, C.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

Wang, D.

Wang, H.

Wang, M. R.

Z. Liu, A. Flores, M. R. Wang, and J. J. Yang, “Diffractive infrared lens with extended depth of focus,” Opt. Eng. 46(1), 018002 (2007).
[Crossref]

Welford, W. T.

Wilson, T.

R. Juškaitis, E. J. Botcherby, and T. Wilson, “Scanning microscopy with extended depth of focus,” Proc. SPIE 5701, 85–92 (2005).
[Crossref]

Xu, Y.

Yamanaka, R.

H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31(Part 1, No. 12B), 4126–4130 (1992).
[Crossref]

Yang, J. J.

Z. Liu, A. Flores, M. R. Wang, and J. J. Yang, “Diffractive infrared lens with extended depth of focus,” Opt. Eng. 46(1), 018002 (2007).
[Crossref]

Ye, R.

Yzuel, M. J.

S. Ledesma, J. C. Escalera, J. Campos, J. Mazzaferri, and M. J. Yzuel, “High depth of focus by combining annular lenses,” Opt. Commun. 266(1), 6–12 (2006).
[Crossref]

R. Hild, M. J. Yzuel, and J. C. Escalera, “High focal depth imaging of small structures,” Microelectron. Eng. 34(2), 195–214 (1997).
[Crossref]

Zhang, H.

Zhou, F.

Appl. Opt. (4)

Atti Fond. Giorgio Ronchi (1)

G. Toraldo di Francia, “Nuovo pupille superresolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).

J. Mod. Opt. (2)

A. Kołodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, “The light sword optical element-a new diffraction structure with extended depth of focus,” J. Mod. Opt. 37(8), 1283–1286 (1990).
[Crossref]

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Calculation of the focusators into a longitudinal line segment and study of a focal area,” J. Mod. Opt. 40(5), 761–769 (1993).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys. (1)

H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31(Part 1, No. 12B), 4126–4130 (1992).
[Crossref]

Microelectron. Eng. (1)

R. Hild, M. J. Yzuel, and J. C. Escalera, “High focal depth imaging of small structures,” Microelectron. Eng. 34(2), 195–214 (1997).
[Crossref]

Opt. Commun. (2)

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Two-zone pupil filters,” Opt. Commun. 281(5), 913–922 (2008).
[Crossref]

S. Ledesma, J. C. Escalera, J. Campos, J. Mazzaferri, and M. J. Yzuel, “High depth of focus by combining annular lenses,” Opt. Commun. 266(1), 6–12 (2006).
[Crossref]

Opt. Eng. (1)

Z. Liu, A. Flores, M. R. Wang, and J. J. Yang, “Diffractive infrared lens with extended depth of focus,” Opt. Eng. 46(1), 018002 (2007).
[Crossref]

Opt. Express (4)

Proc. SPIE (1)

R. Juškaitis, E. J. Botcherby, and T. Wilson, “Scanning microscopy with extended depth of focus,” Proc. SPIE 5701, 85–92 (2005).
[Crossref]

Pure Appl. Opt. (1)

S. Yu. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7(3), 537–548 (1998).
[Crossref]

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

Sov. Tech. Phys. Lett. (1)

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized holograms,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

Other (1)

M. Born, and E. Wolf, Principles of optics, 7th ed., (Cambridge Univ. Press, Cambridge, 1999).

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Figures (8)

Fig. 1
Fig. 1

(Color online) Schematic diagram of an optical system using SWF as the incident wave front modulated element.

Fig. 2
Fig. 2

(Color online) Structure of SWF4 consisting of four CSACSs, marked by k = 0, 1, 2, and 3, respectively (top view). ε represents the normalized radius of center-shaded circle. The inset at the right bottom shows one CSACS of k = 0, and the parabolic shape in the radial direction is represented by gradual changing color.

Fig. 3
Fig. 3

(Color online) Comparison of on-axis intensity profiles before and after optimization design of SWF2 and SWF5 with the same center-shaded radius ε = 0.2 (a). (b) shows the three-dimensional phase profile of SWF5 after optimization, where different colors represent different amplitude transmittances. The effect of burr-like phase relief defect on actual on-axis intensity profile is simulated by adding a normally distributed noise with mean zero and standard deviation σ = std × 2π to the ideal phase profile for SWF2 (c) and SWF5 (d), respectively. Black solid lines represent the ideal intensity profiles for ideal SWF2 and SWF5 without fabrication errors.

Fig. 4
Fig. 4

(Color online) Normalized on-axis intensity profiles after optimization design of SWF2, SWF3, SWF5, and SWF7 with the same center-shaded radius ε = 0.2. The on-axis intensity profile for clear pupil is partially plotted for comparison.

Fig. 5
Fig. 5

(Color online) Normalized transverse intensity profiles for SWF2 in the focal plane. (a) and (b) show the effects of normalized center-shaded radius ε on the transverse intensity distributions for SWF2 along the observation direction φ = 0 and π/8, respectively. (c) and (d) show three-dimensional transverse intensity distributions for clear pupil and SWF2, respectively. (e) Contour of the minimum intensity positions of the transverse main lobe, with v ranging from 3.235 to 3.321 for SWF2 and remaining constant 3.832 for clear pupil.

Fig. 6
Fig. 6

(Color online) Phase profiles of pupil filters: (a) SWF2, (b) annular three-zone binary phase filter, and (c) annular continuous phase filter, respectively (top view).

Fig. 7
Fig. 7

(Color online) Normalized intensity distributions for (a) clear pupil, (b) annular three-zone binary phase filter, and (c) SWF2 at the u-v plane (φ = 0). (d) Normalized on-axis intensity curves with no peak-valley intensity oscillations observed both for annular three-zone binary phase filter and SWF2. (e) Transverse intensity curves for clear pupil, annular three-zone binary phase filter, SWF2 observed at φ = 0 and π/8, respectively.

Fig. 8
Fig. 8

(Color online) Normalized intensity distributions for (a) annular continuous phase filter, and (b) SWF2 at the u-v plane (φ = 0). (c) Normalized on-axis intensity curves with the relative peak-valley intensity oscillations, 43% for annular continuous phase filter and less than 8% for SWF2. (d) The transverse intensity curves for clear pupil, annular continuous phase filter, SWF2 observed at φ = 0 and π/8, respectively.

Tables (1)

Tables Icon

Table 1 Optimized pupil parameters for SWF N (ε = 0.2) calculated by SA

Equations (21)

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

U ( v , φ , u )= 0 1 0 2 π P ( ρ , θ ) exp { j [ v ρ cos ( θ φ ) + u ρ 2 / 2 ] } d θ ρ d ρ ,
P k ( ρ , θ ) = A k exp [ j ϕ k ( ρ , θ ) ] ,
ϕ k ( ρ , θ ) = α k ρ 2 / 2 + β k ,
U swf ( v , φ , u ) = k = 0 N 1 n = 0 3 ε 1 π k / 2 N + n π / 2 π ( k + 1 ) / 2 N + n π / 2 A k exp { j [ ( α k u ) ρ 2 / 2 v ρ cos ( θ φ ) + β k ] } d θ ρ d ρ .
U swf ( u ) = π ( 1 ε 2 ) N k = 0 N 1 A k sinc [ ( 1 ε 2 ) ( u α k ) / 4 π ] exp { j [ ( α k u ) ( 1 + ε 2 ) / 4 + β k ] } ,
U swf ( v , φ ) = 2 k = 0 N 1 ε 1 A k exp [ j ( α k ρ 2 / 2 + β k ) ] π k / 2 N π ( k + 1 ) / 2 N { cos [ v ρ cos ( θ φ ) ] + cos [ v ρ sin ( θ φ ) ] } d θ ρ d ρ .
U swf ( u ) = π ( 1 ε 2 ) N exp [ j u ( 1 + ε 2 ) / 4 ] k = 0 N 1 A k sinc [ ( 1 ε 2 ) ( u α k ) / 4 π ] .
α k = 4 η π 1 ε 2 ( k N 1 2 ) ,
β k = α k ( 1 + ε 2 ) 4 = η π 1 + ε 2 1 ε 2 ( N 1 2 k ) ,
P k ( ρ , θ ) = A k exp { j [ 2 η π 1 ε 2 ( k N 1 2 ) ρ 2 + η π 1 + ε 2 1 ε 2 ( N 1 2 k ) ] } ,
I swf ( v , π / 2 ± φ ) = I swf ( v , φ ) .
I sg ( u ) = I sgmax exp [ ( u 2 / r sg 2 ) N sg ] ,
Minimize δ ( A , η ) = { u [ I swf ( u ) I sg ( u ) ] 2 d u u I sg 2 ( u ) d u } 1 / 2 Subject to: A k = A N 1 k 0.9 A k 1 η > 0
P ( ρ ) = { 0 , 0 < ρ 0.5 exp [ j 4.4 π ( ρ 2 0.5 ) ] , 0.5 < ρ 0.75 . exp [ j 4.4 π ( ρ 2 1 ) ] , 0.75 < ρ 1
U s w f ( u ) = 4 k = 0 N 1 ε 1 π k / 2 N π ( k + 1 ) / 2 N A k exp { j [ ( α k u ) ρ 2 / 2 + β k ] } d θ ρ d ρ = 2 π N k = 0 N 1 ε 1 A k exp { j [ ( α k u ) ρ 2 / 2 + β k ] } ρ d ρ = 2 π N k = 0 N 1 A k exp ( j β k ) 1 j ( α k u ) { exp [ j ( α k u ) / 2 ] exp [ j ( α k u ) ε 2 / 2 ] } = π ( 1 ε 2 ) N k = 0 N 1 A k sinc [ ( u α k ) ( 1 ε 2 ) / ( 4 π ) ] exp { j [ ( α k u ) ( 1 + ε 2 ) / 4 + β k ] } .
U swf ( v , φ ) = k = 0 N 1 n = 0 3 ε 1 A k exp [ j ( α k ρ 2 / 2 + β k ) ] π k / 2 N + n π / 2 π ( k + 1 ) / 2 N + n π / 2 exp [ j v ρ cos ( θ φ ) ] d θ ρ d ρ = k = 0 N 1 ε 1 A k e j ( α k ρ 2 / 2 + β k ) π k / 2 N π ( k + 1 ) / 2 N [ e j v ρ cos ( θ φ ) + e j v ρ sin ( θ φ ) + e j v ρ cos ( θ φ ) + e j v ρ sin ( θ φ ) ] d θ ρ d ρ = 2 k = 0 N 1 ε 1 A k e j ( α k ρ 2 / 2 + β k ) π k / 2 N π ( k + 1 ) / 2 N { cos [ v ρ cos ( θ φ ) ] + cos [ v ρ sin ( θ φ ) ] } d θ ρ d ρ .
U swf ( v , φ + π / 2 ) = U swf ( v , φ ) .
f k ( v ρ , φ ) = π k / 2 N π ( k + 1 ) / 2 N { cos [ v ρ cos ( θ φ ) ] + cos [ v ρ sin ( θ φ ) ] } d θ ,
U swf ( v , φ ) = 2 k = 0 N 1 ε 1 A k e j ( α k ρ 2 / 2 + β k ) f k ( v ρ , φ ) ρ d ρ = 2 k = 0 N / 2 1 [ ε 1 A k e j ( α k ρ 2 / 2 + β k ) f k ( v ρ , φ ) ρ d ρ + ε 1 A N k 1 e j ( α N k 1 ρ 2 / 2 + β N k 1 ) f N k 1 ( v ρ , φ ) ρ d ρ ] = 2 k = 0 N / 2 1 [ ε 1 A k e j ( α k ρ 2 / 2 + β k ) f k ( v ρ , φ ) ρ d ρ + ε 1 A k e j ( α k ρ 2 / 2 + β k ) f N k 1 ( v ρ , φ ) ρ d ρ ] .
U swf ( v , π / 2 φ ) = 2 k = 0 N 1 ε 1 A k e j ( α k ρ 2 / 2 + β k ) π k / 2 N π ( k + 1 ) / 2 N { cos [ v ρ cos ( θ π / 2 + φ ) ] + cos [ v ρ sin ( θ π / 2 + φ ) ] } d θ ρ d ρ = 2 k = 0 N 1 ε 1 A k e j ( α k ρ 2 / 2 + β k ) π / 2 π ( k + 1 ) / 2 N π / 2 π k / 2 N { cos [ v ρ cos ( θ φ ) ] + cos [ v ρ sin ( θ φ ) ] } d θ ρ d ρ = 2 k = 0 N 1 ε 1 A k e j ( α k ρ 2 / 2 + β k ) f N k 1 ( v ρ , φ ) ρ d ρ = 2 k = 0 N / 2 1 [ ε 1 A k e j ( α k ρ 2 / 2 + β k ) f N k 1 ( v ρ , φ ) ρ d ρ + ε 1 A N k 1 e j ( α N k 1 ρ 2 / 2 + β N k 1 ) f k ( v ρ , φ ) ρ d ρ ] = 2 k = 0 N / 2 1 [ ε 1 A k e j ( α k ρ 2 / 2 + β k ) f N k 1 ( v ρ , φ ) ρ d ρ + ε 1 A k e j ( α k ρ 2 / 2 + β k ) f k ( v ρ , φ ) ρ d ρ ] = 2 k = 0 N / 2 1 { [ ε 1 A k e j ( α k ρ 2 / 2 + β k ) f N k 1 ( v ρ , φ ) ρ d ρ ] * + [ ε 1 A k e j ( α k ρ 2 / 2 + β k ) f k ( v ρ , φ ) ρ d ρ ] * } = U swf * ( v , φ ) ,
I swf ( v , π / 2 ± φ ) = I swf ( v , φ ) .

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