Abstract

We suggest a new phase mask coronagraph that can work in a wide band of wavelengths. The phase mask has alternatively sinusoidal and uniform functions in the angular direction. We compare it with the four-quadrant phase mask coronagraph and vortex phase mask coronagraph. Through numerical tests, we find that the new mask gives a deep extinction of star light and has a small inner working angle. It is also shown that this mask has a better performance in chromatism than the others for a wide band of wavelengths.

© 2012 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronagraph: laboratory results and first light at Palomar observatory,” Astrophys. J. 709(1), 53–57 (2010).
[CrossRef]

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

2009

G. A. Swartzlander., “The optical vortex coronagraph,” J. Opt. A, Pure Appl. Opt. 11(9), 094022 (2009).
[CrossRef]

A. Carlotti, G. Ricort, and C. Aime, “Phase mask coronagraphy using a Mach-Zehnder interferometer,” Astron. Astrophys. 504(2), 663–671 (2009).
[CrossRef]

M. P. Cagigal, V. F. Canales, P. J. Valle, and J. E. Oti, “Coronagraphic mask design using Hermite functions,” Opt. Express 17(22), 20515–20520 (2009).
[CrossRef] [PubMed]

2008

G. A. Swartzlander, E. L. Ford, R. S. Abdul-Malik, L. M. Close, M. A. Peters, D. M. Palacios, and D. W. Wilson, “Astronomical demonstration of an optical vortex coronagraph,” Opt. Express 16(14), 10200–10207 (2008).
[CrossRef] [PubMed]

N. Murakami, R. Uemura, N. Baba, J. Nishikawa, M. Tamura, N. Hashimoto, and L. Abe, “An eight-octant phase mask coronagraph,” Publ Astron. Soc. Pac. 120(872), 1112–1118 (2008).
[CrossRef]

2007

J. T. Trauger and W. A. Traub, “A laboratory demonstration of the capability to image an Earth-like extrasolar planet,” Nature 446(7137), 771–773 (2007).
[CrossRef] [PubMed]

D. Rouan, J. Baudrand, A. Boccaletti, P. Baudoz, D. Mawet, and P. Riaud, “The four quadrant phase mask coronagraph and its avatars,” C. R. Phys. 8(3-4), 298–311 (2007).
[CrossRef]

2006

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
[CrossRef] [PubMed]

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. 167(1Suppl.), 81–99 (2006).
[CrossRef]

O. Guyon and M. Shao, “The pupil-swapping coronagraph,” Publ Astron. Soc. Pac. 118(844), 860–865 (2006).
[CrossRef]

G. A. Swartzlander., “Achromatic optical vortex lens,” Opt. Lett. 31(13), 2042–2044 (2006).
[CrossRef] [PubMed]

2005

M. J. Kuchner, J. Crepp, and J. Ge, “Eight-order image masks for terrestrial planet finding,” Astrophys. J. 628(1), 466–473 (2005).
[CrossRef]

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633(2), 1191–1200 (2005).
[CrossRef]

G. Foo, D. M. Palacios, and G. A. Swartzlander., “Optical vortex coronagraph,” Opt. Lett. 30(24), 3308–3310 (2005).
[CrossRef] [PubMed]

2004

2003

O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. 404(1), 379–387 (2003).
[CrossRef]

2002

M. J. Kuchner and W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570(2), 900–908 (2002).
[CrossRef]

N. Baba, N. Murakami, T. Ishigaki, and N. Hashimoto, “Polarization interferometric stellar coronagraph,” Opt. Lett. 27(16), 1373–1375 (2002).
[CrossRef] [PubMed]

2001

L. Abe, F. Vakili, and A. Boccaletti, “The achromatic phase knife coronagraph,” Astron. Astrophys. 374(3), 1161–1168 (2001).
[CrossRef]

2000

D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph,” Publ. Astron. Soc. Pac. 112(777), 1479–1486 (2000).
[CrossRef]

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraph,” Astron. Astrophys. Suppl. Ser. 141(2), 319–329 (2000).
[CrossRef]

1970

1939

M. B. Lyot, “A study of the solar corona and prominences without eclipses,” Mon. Not. R. Astron. Soc. 99, 580–594 (1939).

Abdul-Malik, R. S.

Abe, L.

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

N. Murakami, R. Uemura, N. Baba, J. Nishikawa, M. Tamura, N. Hashimoto, and L. Abe, “An eight-octant phase mask coronagraph,” Publ Astron. Soc. Pac. 120(872), 1112–1118 (2008).
[CrossRef]

L. Abe, F. Vakili, and A. Boccaletti, “The achromatic phase knife coronagraph,” Astron. Astrophys. 374(3), 1161–1168 (2001).
[CrossRef]

Absil, O.

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633(2), 1191–1200 (2005).
[CrossRef]

Aime, C.

A. Carlotti, G. Ricort, and C. Aime, “Phase mask coronagraphy using a Mach-Zehnder interferometer,” Astron. Astrophys. 504(2), 663–671 (2009).
[CrossRef]

Baba, N.

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

N. Murakami, R. Uemura, N. Baba, J. Nishikawa, M. Tamura, N. Hashimoto, and L. Abe, “An eight-octant phase mask coronagraph,” Publ Astron. Soc. Pac. 120(872), 1112–1118 (2008).
[CrossRef]

N. Baba, N. Murakami, T. Ishigaki, and N. Hashimoto, “Polarization interferometric stellar coronagraph,” Opt. Lett. 27(16), 1373–1375 (2002).
[CrossRef] [PubMed]

Baudoz, P.

D. Rouan, J. Baudrand, A. Boccaletti, P. Baudoz, D. Mawet, and P. Riaud, “The four quadrant phase mask coronagraph and its avatars,” C. R. Phys. 8(3-4), 298–311 (2007).
[CrossRef]

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraph,” Astron. Astrophys. Suppl. Ser. 141(2), 319–329 (2000).
[CrossRef]

Baudrand, J.

D. Rouan, J. Baudrand, A. Boccaletti, P. Baudoz, D. Mawet, and P. Riaud, “The four quadrant phase mask coronagraph and its avatars,” C. R. Phys. 8(3-4), 298–311 (2007).
[CrossRef]

Boccaletti, A.

D. Rouan, J. Baudrand, A. Boccaletti, P. Baudoz, D. Mawet, and P. Riaud, “The four quadrant phase mask coronagraph and its avatars,” C. R. Phys. 8(3-4), 298–311 (2007).
[CrossRef]

L. Abe, F. Vakili, and A. Boccaletti, “The achromatic phase knife coronagraph,” Astron. Astrophys. 374(3), 1161–1168 (2001).
[CrossRef]

D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph,” Publ. Astron. Soc. Pac. 112(777), 1479–1486 (2000).
[CrossRef]

Burruss, R.

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronagraph: laboratory results and first light at Palomar observatory,” Astrophys. J. 709(1), 53–57 (2010).
[CrossRef]

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

Cagigal, M. P.

Canales, V. F.

Cao, Q.

Carlotti, A.

A. Carlotti, G. Ricort, and C. Aime, “Phase mask coronagraphy using a Mach-Zehnder interferometer,” Astron. Astrophys. 504(2), 663–671 (2009).
[CrossRef]

Clénet, Y.

D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph,” Publ. Astron. Soc. Pac. 112(777), 1479–1486 (2000).
[CrossRef]

Close, L. M.

Collins, B.

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. 167(1Suppl.), 81–99 (2006).
[CrossRef]

Collins, S. A.

Crepp, J.

M. J. Kuchner, J. Crepp, and J. Ge, “Eight-order image masks for terrestrial planet finding,” Astrophys. J. 628(1), 466–473 (2005).
[CrossRef]

Foo, G.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
[CrossRef] [PubMed]

G. Foo, D. M. Palacios, and G. A. Swartzlander., “Optical vortex coronagraph,” Opt. Lett. 30(24), 3308–3310 (2005).
[CrossRef] [PubMed]

Ford, E. L.

Gay, J.

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraph,” Astron. Astrophys. Suppl. Ser. 141(2), 319–329 (2000).
[CrossRef]

Ge, J.

M. J. Kuchner, J. Crepp, and J. Ge, “Eight-order image masks for terrestrial planet finding,” Astrophys. J. 628(1), 466–473 (2005).
[CrossRef]

Gruber, M.

Guyon, O.

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. 167(1Suppl.), 81–99 (2006).
[CrossRef]

O. Guyon and M. Shao, “The pupil-swapping coronagraph,” Publ Astron. Soc. Pac. 118(844), 860–865 (2006).
[CrossRef]

O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. 404(1), 379–387 (2003).
[CrossRef]

Hashimoto, N.

N. Murakami, R. Uemura, N. Baba, J. Nishikawa, M. Tamura, N. Hashimoto, and L. Abe, “An eight-octant phase mask coronagraph,” Publ Astron. Soc. Pac. 120(872), 1112–1118 (2008).
[CrossRef]

N. Baba, N. Murakami, T. Ishigaki, and N. Hashimoto, “Polarization interferometric stellar coronagraph,” Opt. Lett. 27(16), 1373–1375 (2002).
[CrossRef] [PubMed]

Hickey, J.

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronagraph: laboratory results and first light at Palomar observatory,” Astrophys. J. 709(1), 53–57 (2010).
[CrossRef]

Ishigaki, T.

Jahns, J.

Johnson, E. G.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
[CrossRef] [PubMed]

Kuchner, M. J.

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. 167(1Suppl.), 81–99 (2006).
[CrossRef]

M. J. Kuchner, J. Crepp, and J. Ge, “Eight-order image masks for terrestrial planet finding,” Astrophys. J. 628(1), 466–473 (2005).
[CrossRef]

M. J. Kuchner and W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570(2), 900–908 (2002).
[CrossRef]

Labeyrie, A.

D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph,” Publ. Astron. Soc. Pac. 112(777), 1479–1486 (2000).
[CrossRef]

Lee, J. H.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
[CrossRef] [PubMed]

Liewer, K.

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronagraph: laboratory results and first light at Palomar observatory,” Astrophys. J. 709(1), 53–57 (2010).
[CrossRef]

Lyot, M. B.

M. B. Lyot, “A study of the solar corona and prominences without eclipses,” Mon. Not. R. Astron. Soc. 99, 580–594 (1939).

Mawet, D.

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronagraph: laboratory results and first light at Palomar observatory,” Astrophys. J. 709(1), 53–57 (2010).
[CrossRef]

D. Rouan, J. Baudrand, A. Boccaletti, P. Baudoz, D. Mawet, and P. Riaud, “The four quadrant phase mask coronagraph and its avatars,” C. R. Phys. 8(3-4), 298–311 (2007).
[CrossRef]

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633(2), 1191–1200 (2005).
[CrossRef]

Murakami, N.

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

N. Murakami, R. Uemura, N. Baba, J. Nishikawa, M. Tamura, N. Hashimoto, and L. Abe, “An eight-octant phase mask coronagraph,” Publ Astron. Soc. Pac. 120(872), 1112–1118 (2008).
[CrossRef]

N. Baba, N. Murakami, T. Ishigaki, and N. Hashimoto, “Polarization interferometric stellar coronagraph,” Opt. Lett. 27(16), 1373–1375 (2002).
[CrossRef] [PubMed]

Nishikawa, J.

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

N. Murakami, R. Uemura, N. Baba, J. Nishikawa, M. Tamura, N. Hashimoto, and L. Abe, “An eight-octant phase mask coronagraph,” Publ Astron. Soc. Pac. 120(872), 1112–1118 (2008).
[CrossRef]

Oti, J. E.

Palacios, D. M.

Peters, M. A.

Pluzhnik, E. A.

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. 167(1Suppl.), 81–99 (2006).
[CrossRef]

Rabbia, Y.

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraph,” Astron. Astrophys. Suppl. Ser. 141(2), 319–329 (2000).
[CrossRef]

Riaud, P.

D. Rouan, J. Baudrand, A. Boccaletti, P. Baudoz, D. Mawet, and P. Riaud, “The four quadrant phase mask coronagraph and its avatars,” C. R. Phys. 8(3-4), 298–311 (2007).
[CrossRef]

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633(2), 1191–1200 (2005).
[CrossRef]

D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph,” Publ. Astron. Soc. Pac. 112(777), 1479–1486 (2000).
[CrossRef]

Ricort, G.

A. Carlotti, G. Ricort, and C. Aime, “Phase mask coronagraphy using a Mach-Zehnder interferometer,” Astron. Astrophys. 504(2), 663–671 (2009).
[CrossRef]

Ridgway, S. T.

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. 167(1Suppl.), 81–99 (2006).
[CrossRef]

Rouan, D.

D. Rouan, J. Baudrand, A. Boccaletti, P. Baudoz, D. Mawet, and P. Riaud, “The four quadrant phase mask coronagraph and its avatars,” C. R. Phys. 8(3-4), 298–311 (2007).
[CrossRef]

D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph,” Publ. Astron. Soc. Pac. 112(777), 1479–1486 (2000).
[CrossRef]

Serabyn, E.

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronagraph: laboratory results and first light at Palomar observatory,” Astrophys. J. 709(1), 53–57 (2010).
[CrossRef]

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

Shao, M.

O. Guyon and M. Shao, “The pupil-swapping coronagraph,” Publ Astron. Soc. Pac. 118(844), 860–865 (2006).
[CrossRef]

Shemo, D.

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronagraph: laboratory results and first light at Palomar observatory,” Astrophys. J. 709(1), 53–57 (2010).
[CrossRef]

Surdej, J.

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633(2), 1191–1200 (2005).
[CrossRef]

Swartzlander, G. A.

Tamura, M.

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

N. Murakami, R. Uemura, N. Baba, J. Nishikawa, M. Tamura, N. Hashimoto, and L. Abe, “An eight-octant phase mask coronagraph,” Publ Astron. Soc. Pac. 120(872), 1112–1118 (2008).
[CrossRef]

Traub, W. A.

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J. T. Trauger and W. A. Traub, “A laboratory demonstration of the capability to image an Earth-like extrasolar planet,” Nature 446(7137), 771–773 (2007).
[CrossRef] [PubMed]

Uemura, R.

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N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
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Figures (5)

Fig. 1
Fig. 1

Set-up of the proposed coronagraphic system; it is composed of three imaging lenses (L1, L2, L3), aperture stop (AS), Lyot stop (LS), the mask, and an detecting system like CCD camera. All lenses have the same focal lengths f.

Fig. 2
Fig. 2

(a) Surface profile of the SPM composed of sinusoidal region and flat region. With the thickness of mask increasing, the color changes from blue to red. (b) The corresponding phase distribution in the angular direction for the designed wavelength λ0. Note that the function G(θ) is a double periodic function in the range of 0<θ≤2π. Also note that the unit of h is radian.

Fig. 3
Fig. 3

Acquired Lyot-stop image of |U(x′′,y′′)|2. (a) The calculation by use of Eqs. (5) and (15). (b) The simulation by use of 2-D FFT. The color represents relative intensity. When the value of relative intensity goes from 0 to 1, the color will change from blue to red. The diameter of the “dark” circular region in the middle is 1.

Fig. 4
Fig. 4

Comparison between the peak throughput of SPM (blue curve), FQPM (green curve), and VPM2 (red curve), for RLS = RAS. It is pointed out that the pitch multiplicity of VPM2 is chosen equal to two [23], and d is the diameter of aperture stop, where d = 2RAS.

Fig. 5
Fig. 5

Comparison among the value |C0(λ)|2 of SPM (blue curve), FQPM (green curve), and VPM2 (red curve), for designed wavelength λ0 = 550nm.

Equations (33)

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circ(r/ R AS )={ 1 r R AS 0 r> R AS ,
U(x',y')=t(r',θ) (i λ 0 f) 1 exp(ik r ' 2 / 2f )2π A 0 R AS 2 J 1 (ar') ar' ,
U(x',y')= A 1 t(θ)exp(ikr ' 2 /2f) J 1 (ar') ar' ,
U(x'',y'')= A 2 n= (i) n C n H n { J 1 (ar') ar' } e inφ ,
C n = 1 2π 0 2π t(θ) e inθ dθ .
n= | C n | 2 =1
H n { J 1 (ar') ar' }={ f n (r''), r'' R AS 0 r''< R AS ,
C 2q+1 (λ)= 1 2π [ 0 π t(θ,λ) e i(2q+1)θ dθ + π 2π t(θ,λ) e i(2q+1)θ dθ ] = 1 2π [ 0 π t(θ,λ) e i(2q+1)θ dθ + e i(2q+1)π 0 π t(θ+π,λ) e i(2q+1)θ dθ ].
C 0 (λ)= C 0 ( λ 0 )+ C 0 (λ) λ | λ 0 (λ λ 0 )+ 1 2 2 C 0 (λ) λ 2 | λ 0 (λ λ 0 ) 2 +.
{ C 0 ( λ 0 )=0 C 0 (λ) λ | λ 0 =0.
t(θ,λ)={ e i λ 0 λ hsin(bθ) 0θ< 2π b 1 2π b θ<π, 2π b +πθ<2π e i λ 0 λ hsin[ b(θπ) ] πθ< 2π b +π ,
{ J 0 (x)= 1 2π 0 2π e ixsinθ dθ J 0 '(x)=J (x) 1 ,
{ J 0 (h)=1 b 2 J 1 (h)=0 .
C 0 (λ)| SPM = 2 b J 0 ( λ 0 λ h )+1 2 b .
U(x'',y'')| SPM = A 2 m= (i) 2m C 2m H 2m { J 1 (ar') ar' } e i2mφ ,
m=15 15 | C 2m | 2 =0.9996.
{ | C 0 (λ) | FQPM 2 = cos 2 ( λ 0 2λ π ) | C 0 (λ) | VPM2 2 = sin 2 ( λ 0 λ π ) ( λ 0 λ π ) 2 .
[ A B C D ],
M 1 =[ 1 f 0 1 ][ 1 0 f 1 1 ]=[ 0 f f 1 1 ].
U(x',y')=t(r',θ) (iλB) 1 exp(ikf) A 0 circ(r/ R AS ) ×exp{ ik 2B [ A( x 2 + y 2 )+D( x ' 2 +y ' 2 )2( xx'+yy' ) ] }dxdy.
{ x=rcosφ x'=rcosθ y=rsinφ y'=rsinθ
U(x',y')=t(r',θ) (iλf) 1 exp( ikr ' 2 / 2f ) × A 0 circ(r/ R AS ) exp[ ik f (xx'+yy')]dxdy =t(r',θ) (iλf) 1 exp( ikr ' 2 / 2f ) × 0 A 0 rcirc(r/ R AS ){ 0 2π exp[ ik f rr'cos(φθ)]dφ }dr .
{ exp[ i2πrρcos(φθ) ]= k= (i) k J k (2πrρ)exp[ ik(φθ) ] 0 ρ r J 0 (r)dr =ρ J 1 (ρ) ,
U(x',y')=t(r',θ) (iλf) 1 exp( ikr ' 2 / 2f ) × n= 0 A 0 rcirc(r/ R AS ) J n ( k f rr' ){ 0 2π (i) n exp[ in(φθ) ]dφ }dr =t(r',θ) (iλf) 1 exp( ikr ' 2 / 2f ) × n= 0 A 0 rcirc(r/ R AS ) J n ( k f rr' ) (i) n exp(inθ)[ 0 2π exp(inφ)dφ ]dr =t(r',θ) (iλf) 1 exp( ikr ' 2 / 2f )2π A 0 0 rcirc(r/ R AS ) J 0 ( k f r'r)dr =t(r',θ) (i λ 0 f) 1 exp(ik r ' 2 / 2f )2π A 0 R AS 2 J 1 ( k R AS f r') k R AS f r' .
M 2 =[ 1 2f 0 1 ][ 1 0 f 1 1 ][ 1 f 0 1 ]=[ 1 f f 1 0 ].
U(x'',y'')= (iλB) 1 exp(i3kf) U(x',y') ×exp{ ik 2B [ A( x ' 2 +y ' 2 )+D( x' ' 2 +y' ' 2 )2( x'x''+y'y'' ) ] }dx'dy'.
U(x'',y'')= (iλf) 1 A 1 t(θ) J 1 (ar') ar' exp(ikr ' 2 /2f) ×exp{ ik 2f [ ( x ' 2 +y ' 2 )2( x'x''+y'y'' ) ] }dx'dy' = A 1 (iλf) 1 t(θ) J 1 (ar') ar' exp[ ik f ( x'x''+y'y'' )]dx'dy'.
U(x'',y'')= A 1 (iλf) 1 0 r' J 1 (ar') ar' { 0 2π t(θ)exp[ ik f r'r''cos(θϕ)]dθ }dr' = A 1 (iλf) 1 n= 0 r' J 1 (ar') ar' J n ( k f r'r'') ×{ 0 2π (i) n t(θ) exp[ ik(θϕ) ]dθ }dr' = A 1 (iλf) 1 n= 0 r' J 1 (ar') ar' J n ( k f r'r'') × (i) n exp(inϕ){ 0 2π t(θ) exp(inθ)dθ }dr' = A 2 n= (i) n C n H n { J 1 (ar') ar' } exp(inϕ),
H 2m { J 1 (r') r' }= 0 J 2m (r''r') J 1 (r')dr ' = Γ(m+1) r' ' 2 Γ(2)Γ(m) × F 2 1 (1+m,1m;2; 1 r' ' 2 ), r''1
H 2m { J 1 (r') r' }= m r' ' 2 n=0 m1 (1+m) n (1m) n 2 n n! 1 r' ' 2n ,
(x) n =x(x+1)(x+2)(x+n1).
H 2m { J 1 (r') r' }= m r' ' 2 n=0 m1 Γ(m+n+1) Γ(1+m) (1) n Γ(m) Γ(mn) Γ(n+2)Γ(n+1) 1 r' ' 2n = m r' ' 2 n=0 m1 (1) n Γ(m+n+1)Γ(m) Γ(1+m)Γ(mn)Γ(n+2)Γ(n+1) 1 r' ' 2n = m r' ' 2 n=0 m1 (1) n (m+n)!(m1)! m!n!(mn1)!(n+1)! 1 r' ' 2n ,
H 2 { J 1 (r') r' }= 1 r' ' 2 H 4 { J 1 (r') r' }= 2 r' ' 2 + 3 r' ' 4 H 6 { J 1 (r') r' }= 3 r' ' 2 12 r' ' 4 + 10 r' ' 6 .

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