Abstract

The isotropic and anisotropic image edge enhancements by employing Airy spiral phase filters are proposed and demonstrated. The coherent spread functions of the image systems are derived from transmittance functions of their corresponding filters. In the isotropic method, the distributions of the coherent spread function with the radius of the main ring ρ0 and the scaled parameter w0 are numerically analyzed. It is found that the width of the main lobe determining the resolution decreases with the increased ρ0, and the amplitudes of the side lobes connecting with the contrast fluctuate with w0. Compared with the existing spiral phase filters, higher contrast and resolution can be achieved by adjusting the two parameters in the Airy spiral phase filter. Moreover, an off-axis Airy spiral phase filter by controlling the center position (ρ0,ϕ1) is designed and employed to implement anisotropic edge enhancement. In the experiments, two methods of image edge enhancement have been verified by using the amplitude-contrast and phase-contrast objects.

© 2016 Optical Society of America

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References

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2016 (1)

B. Zhang, Z. Chen, H. Sun, J. Xia, and J. Ding, “Vectorial optical vortex filtering for edge Enhancement,” J. Opt. 18(3), 035703 (2016).
[Crossref]

2015 (1)

J. Wang, W. Zhang, Q. Qi, S. Zheng, and L. Chen, “Gradual edge enhancement in spiral phase contrast imaging with fractional vortex filters,” Sci. Rep. 5, 15826 (2015).
[Crossref] [PubMed]

2014 (2)

Z. Liu, L. Tian, S. Liu, and L. Waller, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19(10), 106002 (2014).
[Crossref] [PubMed]

M. K. Sharma, J. Joseph, and P. Senthilkumaran, “Directional edge enhancement using superposed vortex filter,” Opt. Laser Technol. 57, 230–235 (2014).
[Crossref]

2013 (2)

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

M. K. Sharma, J. Joseph, and P. Senthilkumaran, “Selective edge enhancement using shifted anisotropic vortex filter,” J. Opt. 42(1), 1–7 (2013).
[Crossref]

2011 (3)

2010 (4)

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283(7), 1273–1287 (2010).
[Crossref]

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010).
[Crossref] [PubMed]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref] [PubMed]

N. Zhang, J. A. Davis, I. Moreno, J. Lin, K. Moh, D. M. Cottrell, and X.-C. Yuan, “Analysis of fractional vortex beams using a vortex grating spectrum analyzer,” Appl. Opt. 49(13), 2456–2462 (2010).
[Crossref]

2009 (2)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy Beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

G. Situ, G. Pedrini, and W. Osten, “Spiral phase filtering and orientation-selective edge detection/enhancement,” J. Opt. Soc. Am. A 26(8), 1788–1797 (2009).
[Crossref] [PubMed]

2008 (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” nat. Photonics 2(11), 675–678 (2008).
[Crossref]

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (3)

2004 (1)

2000 (1)

1992 (1)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Abdollahpour, D.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref] [PubMed]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Bech, M.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Bernet, S.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005).
[Crossref] [PubMed]

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Bruyndonckx, P.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Campos, J.

Chen, L.

J. Wang, W. Zhang, Q. Qi, S. Zheng, and L. Chen, “Gradual edge enhancement in spiral phase contrast imaging with fractional vortex filters,” Sci. Rep. 5, 15826 (2015).
[Crossref] [PubMed]

Chen, Z.

Christodoulides, D. N.

Cottrell, D. M.

Crabtree, K.

Davis, J. A.

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Ding, J.

B. Zhang, Z. Chen, H. Sun, J. Xia, and J. Ding, “Vectorial optical vortex filtering for edge Enhancement,” J. Opt. 18(3), 035703 (2016).
[Crossref]

C. S. Guo, Y. J. Han, J. B. Xu, and J. Ding, “Radial Hilbert transform with Laguerre-Gaussian spatial filters,” Opt. Lett. 31(10), 1394–1396 (2006).
[Crossref] [PubMed]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Efremidis, N. K.

Foo, G.

Fürhapter, S.

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005).
[Crossref] [PubMed]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005).
[Crossref] [PubMed]

Guo, C. S.

Han, Y. J.

Hostens, J.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Jesacher, A.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005).
[Crossref] [PubMed]

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005).
[Crossref] [PubMed]

Joseph, J.

M. K. Sharma, J. Joseph, and P. Senthilkumaran, “Directional edge enhancement using superposed vortex filter,” Opt. Laser Technol. 57, 230–235 (2014).
[Crossref]

M. K. Sharma, J. Joseph, and P. Senthilkumaran, “Selective edge enhancement using shifted anisotropic vortex filter,” J. Opt. 42(1), 1–7 (2013).
[Crossref]

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy Beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

Lin, J.

Liu, S.

Z. Liu, L. Tian, S. Liu, and L. Waller, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19(10), 106002 (2014).
[Crossref] [PubMed]

Liu, Z.

Z. Liu, L. Tian, S. Liu, and L. Waller, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19(10), 106002 (2014).
[Crossref] [PubMed]

Mawet, D.

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” nat. Photonics 2(11), 675–678 (2008).
[Crossref]

McNamara, D. E.

Mills, M. S.

Moh, K.

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy Beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Moreno, I.

Osten, W.

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283(7), 1273–1287 (2010).
[Crossref]

G. Situ, G. Pedrini, and W. Osten, “Spiral phase filtering and orientation-selective edge detection/enhancement,” J. Opt. Soc. Am. A 26(8), 1788–1797 (2009).
[Crossref] [PubMed]

Palacios, D. M.

Papazoglou, D. G.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref] [PubMed]

Pauwels, B.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Pedrini, G.

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283(7), 1273–1287 (2010).
[Crossref]

G. Situ, G. Pedrini, and W. Osten, “Spiral phase filtering and orientation-selective edge detection/enhancement,” J. Opt. Soc. Am. A 26(8), 1788–1797 (2009).
[Crossref] [PubMed]

Pfeiffer, F.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy Beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Prakash, J.

Pueyo, L.

Qi, Q.

J. Wang, W. Zhang, Q. Qi, S. Zheng, and L. Chen, “Gradual edge enhancement in spiral phase contrast imaging with fractional vortex filters,” Sci. Rep. 5, 15826 (2015).
[Crossref] [PubMed]

Ritsch-Marte, M.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005).
[Crossref] [PubMed]

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005).
[Crossref] [PubMed]

Salandrino, A.

Sasov, A.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Senthilkumaran, P.

M. K. Sharma, J. Joseph, and P. Senthilkumaran, “Directional edge enhancement using superposed vortex filter,” Opt. Laser Technol. 57, 230–235 (2014).
[Crossref]

M. K. Sharma, J. Joseph, and P. Senthilkumaran, “Selective edge enhancement using shifted anisotropic vortex filter,” J. Opt. 42(1), 1–7 (2013).
[Crossref]

Serabyn, E.

Sharma, M. K.

M. K. Sharma, J. Joseph, and P. Senthilkumaran, “Directional edge enhancement using superposed vortex filter,” Opt. Laser Technol. 57, 230–235 (2014).
[Crossref]

M. K. Sharma, J. Joseph, and P. Senthilkumaran, “Selective edge enhancement using shifted anisotropic vortex filter,” J. Opt. 42(1), 1–7 (2013).
[Crossref]

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

Situ, G.

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283(7), 1273–1287 (2010).
[Crossref]

G. Situ, G. Pedrini, and W. Osten, “Spiral phase filtering and orientation-selective edge detection/enhancement,” J. Opt. Soc. Am. A 26(8), 1788–1797 (2009).
[Crossref] [PubMed]

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy Beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

Sun, H.

B. Zhang, Z. Chen, H. Sun, J. Xia, and J. Ding, “Vectorial optical vortex filtering for edge Enhancement,” J. Opt. 18(3), 035703 (2016).
[Crossref]

Suntsov, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref] [PubMed]

Swartzlander, G. A.

Tapfer, A.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Tian, L.

Z. Liu, L. Tian, S. Liu, and L. Waller, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19(10), 106002 (2014).
[Crossref] [PubMed]

Tzortzakis, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref] [PubMed]

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

Velroyen, A.

M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, P. Bruyndonckx, A. Sasov, and F. Pfeiffer, “In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. 3, 3209 (2013).
[Crossref] [PubMed]

Wallace, J. K.

Waller, L.

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Figures (10)

Fig. 1
Fig. 1 (a) The diagram of the imaging system. L1 and L2 represent lenses, f 1 and f 2 are focal length of L1 and L2, respectively. (b) The pattern of the AiSF.
Fig. 2
Fig. 2 The CSF of AiSF with (a) the different values of ρ 0 when w 0 = 0.5 R 0 , (b) the different values of w 0 when ρ 0 = 0.6 R 0 .
Fig. 3
Fig. 3 CSF of: (a) SPP, (b) LGSF (c) BLSF and (d) AiSF, (e)-(h) are amplitude profile in radial section of (a)-(d), respectively.
Fig. 4
Fig. 4 CSF with ( ρ 1 , ϕ 1 ) equal to: (a) ( R 0 /8,0 ) , (b) ( R 0 /4,0 ) , (c) ( R 0 /2,0 ) , (d) ( R 0 ,0 ) , (e) - (h) are amplitude profile in radial section of (a)-(d), respectively.
Fig. 5
Fig. 5 CSF of AiSF with ( ρ 1 , ϕ 1 ) equal to (a) ( R 0 /2,0 ) , (b) ( R 0 /2,π/4 ) , respectively.
Fig. 6
Fig. 6 The first row is the off-axis AiSF with different ( ρ 1 , ϕ 1 ) . The values of ( ρ 1 , ϕ 1 ) are (a1) ( R 0 /8,3π/2 ) , (b1) ( R 0 ,π/2 ) , (c1) ( R 0 /4,0 ) , (d1) ( R 0 /4,π ) . The second row (a2)-(d2) are the output modified images which are modulated by the off-axis filters with corresponding ( ρ 1 , ϕ 1 ) in (a1)-(d1), respectively. The third row (a3)-(d3) are the intensity section distributions of (a2)-(d2) along the red arrow, respectively.
Fig. 7
Fig. 7 (a) The experimental setup for edge enhancement. L2, L3 and L4 are the Fourier lens; BE is beam expander; BS is the beam splitter; SLM is the spatial light modulator; CCD is the Charge Coupled Device camera, (b) is the CGH based on AiSF.
Fig. 8
Fig. 8 The recorded images of an amplitude-contrast object by using (a) conventional bright field, (b) LGSF,(c) BLSF, (d) AiSF, (e)-(h) are the intensity section distribution of (a)-(d), respectively.
Fig. 9
Fig. 9 The images of a phase-contrast object by using (a) phase contrast microscope, (b) LGSF, (c) BLSF, (d) AiSF, respectively.
Fig. 10
Fig. 10 The image of a phase-contrast object by using AiSF with different ( ρ 1 , ϕ 1 ) . The values of ( ρ 1 , ϕ 1 ) are (a) ( R 0 /8,3π/2 ) , (b) ( R 0 ,π/2 ) , (c) ( R 0 /4,0 ) , (d) ( R 0 /4,π ) , respectively. In the below left hand corner, the subfigure is the magnified image of the red box part image. In the below right hand corner, the subfigure shows the cross sections of magnified image along the red arrows.

Equations (11)

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o( r 0 , θ 0 )=| o( r 0 , θ 0 ) |exp[ iΦ( r 0 , θ 0 ) ],
o( r 0 , θ 0 )=| o( r 0 , θ 0 ) |[ 1+iΦ( r 0 , θ 0 )+ [ iΦ( r 0 , θ 0 ) ] 2 2! + [ iΦ( r 0 , θ 0 ) ] 3 3! + ].
O( ρ,ϕ )=F[ o( r 0 , θ 0 ) ]×H( ρ,ϕ ).
H( ρ,ϕ )=Ai( ρ 0 ρ w 0 )exp( ρ 0 ρ w 0 )circl( ρ R 0 )exp( iϕ ),
h( r 1 , θ 1 )= 2π λ f 2 exp( i θ 1 ) 0 R 0 Ai( ρ 0 ρ w 0 ) exp( ρ 0 ρ w 0 ) J 1 ( 2π r 1 ρ λ f 2 )ρdρ,
o ˜ ( r 1 , θ 1 )=o( r 0 , θ 0 )h( r 1 , θ 1 ) = exp[ i4πk( f 1 + f 2 ) ] 0 R 0 0 2π h( r 1 r 0 , θ 1 θ 0 )| o( r 0 , θ 0 ) | [ 1+iΦ( r 0 , θ 0 )+ [ iΦ( r 0 , θ 0 ) ] 2 2! + [ iΦ( r 0 , θ 0 ) ] 3 3! + ]d r 0 d θ 0 ,
o ˜ ( r 1 , θ 1 )=C+ 0 R 0 0 2π h( r 1 r 0 , θ 1 θ 0 ) | o( r 0 , θ 0 ) | { iΦ( r 0 , θ 0 )+ [ iΦ( r 0 , θ 0 ) ] 2 2! + [ iΦ( r 0 , θ 0 ) ] 3 3! + }d r 0 d θ 0 ,
I( r 1 , θ 1 )= | o ˜ ( r 1 , θ 1 ) | 2 .
H off (ρ,ϕ, ρ 1 ,ϕ ) 1 =Ai( B w )×exp( B w )× [ ρexp( iϕ ) ρ 1 exp( i ϕ 1 ) ] C ,
B= ( ρ 0 ρ ) 2 + ρ 1 2 +2( ρ 0 ρ ) ρ 1 cos( ϕ ϕ 1 ) C= ρ 2 + ρ 1 2 +2ρ ρ 1 cos( ϕ ϕ 1 ).
h off (r,φ, ρ 1 , ϕ 1 )=F[ H off (ρ,ϕ, ρ 1 , ϕ 1 ) ].

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