Abstract

We report a detailed investigation on the second harmonic generation (SHG) emission from single 150 nm diameter non-centrosymmetric gold nanoparticles. Polarization-resolved analysis together with scanning electron microscopy images shows that these nanostructures exhibit a unique polarization-sensitive SHG that depends strongly on the particle’s shape. An analytical approach based on multipolar analysis is introduced to link SHG properties to the nanoparticles’ shape. Those multipolar modes can be probed using polarization-resolved SHG. This multipolar analysis offers a physical picture of the relation between shape (size, symmetries, defects, etc.) and nonlinear polarized optical efficiency.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (2)

M. Guasoni, L. Carletti, D. Neshev, and C. De Angelis, “Theoretical model for pattern engineering of harmonic generation in all-dielectric nanoantennas,” IEEE J. Quantum Elect. 53, 1–5 (2017).
[Crossref]

L. Carletti, D. Rocco, A. Locatelli, C. De Angelis, V. Gili, M. Ravaro, I. Favero, G. Leo, M. Finazzi, L. Ghirardini, M. Celebrano, G. Marino, and A. V. Zayats, “Controlling second-harmonic generation at the nanoscale with monolithic algaas-on-alox antennas,” Nanotechnol. 28, 114005 (2017).
[Crossref]

2016 (3)

J. R. Rouxel and T. Toury, “Optical multipolar spread functions of an aplanatic imaging system,” J. Opt. 18, 075002 (2016).
[Crossref]

N. G. Bastús, J. Piella, and V. Puntes, “Quantifying the sensitivity of multipolar (dipolar, quadrupolar, and octapolar) surface plasmon resonances in silver nanoparticles: The effect of size, composition, and surface coating,” Langmuir 32, 290–300 (2016).
[Crossref]

S. D. Gennaro, M. Rahmani, V. Giannini, H. Aouani, T. P. Sidiropoulos, M. Navarro-Cía, S. A. Maier, and R. F. Oulton, “The interplay of symmetry and scattering phase in second harmonic generation from gold nanoantennas,” Nano Lett. 16, 5278–5285 (2016).
[Crossref] [PubMed]

2015 (2)

D. de Ceglia, M. A. Vincenti, C. De Angelis, A. Locatelli, J. W. Haus, and M. Scalora, “Role of antenna modes and field enhancement in second harmonic generation from dipole nanoantennas,” Opt. Express 23, 1715–1729 (2015).
[Crossref] [PubMed]

M. Celebrano, X. Wu, M. Baselli, S. Großmann, P. Biagioni, A. Locatelli, C. De Angelis, G. Cerullo, R. Osellame, B. Hecht, Duò Lamberto, Franco Ciccacci, and Marco Finazzi, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nat. Nanotechnol. 10, 412–417 (2015).
[Crossref] [PubMed]

2013 (1)

2012 (3)

H. Shen, N. Guillot, J. Rouxel, M. Lamy de la Chapelle, and T. Toury, “Optimized plasmonic nanostructures for improved sensing activities,” Opt. Express 20, 21278–21290 (2012).
[Crossref] [PubMed]

H. Shen, J. Rouxel, N. Guillot, M. Lamy de la Chapelle, and T. Toury, “Light polarization properties of three fold symmetry gold nanoparticles: Model and experiments,” C. R. Phys. 13, 830–836 (2012).
[Crossref]

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6, 737–748 (2012).
[Crossref]

2011 (1)

A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Multipole light scattering by nonspherical nanoparticles in the discrete dipole approximation,” Phys. Rev. B 84, 235429 (2011).
[Crossref]

2008 (2)

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials: erratum,” Opt. Express 16, 8055 (2008).
[Crossref]

G. Bachelier, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” Phys. Rev. 25, 955–960 (2008).

2007 (5)

M. Finazzi, P. Biagioni, M. Celebrano, and L. Duo, “Selection rules for second-harmonic generation in nanoparticles,” Phys. Rev. B 76, 125414 (2007).
[Crossref]

C. Hubert, L. Billot, P.-M. Adam, R. Bachelot, P. Royer, J. Grand, D. Gindre, K. D. Dorkenoo, and A. Fort, “Role of surface plasmon in second harmonic generation from gold nanorods,” Appl. Phys. Lett. 90, 181105 (2007).
[Crossref]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
[Crossref] [PubMed]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
[Crossref] [PubMed]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
[Crossref] [PubMed]

2006 (1)

E. Delahaye, N. Tancrez, T. Yi, I. Ledoux, J. Zyss, S. Brasselet, and R. Clément, “Second harmonic generation from individual hybrid MnPS3,” Chem. Phys. Lett. 429, 533–537 (2006).
[Crossref]

2005 (2)

J. Nappa, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Wavelength dependence of the retardation effects in silver nanoparticles followed by polarization resolved hyper rayleigh scattering,” Chem. Phys. Lett. 415, 246–250 (2005).
[Crossref]

J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71, 165407 (2005).
[Crossref]

2004 (2)

J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21, 1328–1347 (2004).
[Crossref]

S. Brasselet, V. L. Floc’h, F. Treussart, J.-F. Roch, and J. Zyss, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[Crossref] [PubMed]

2003 (4)

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

A. Nahata, R. A. Linke, T. Ishi, and K. Ohashi, “Enhanced nonlinear optical conversion from a periodically nanostructured metal film,” Opt. Lett. 28, 423–425 (2003).
[Crossref] [PubMed]

C. I. Valencia and E. R. Méndez, “Second-harmonic generation in the scattering of light by two-dimensional particles,” J. Opt. Soc. Am. B 20, 2150–2161 (2003).
[Crossref]

J. Grand, S. Kostcheev, J.-L. Bijeon, M. Lamy de la Chapelle, P.-M. Adam, A. Rumyantseva, G. Lerondel, and P. Royer, “Optimization of sers-active substrates for near-field raman spectroscopy,” Synth. Met. 139, 621–624 (2003).
[Crossref]

2000 (1)

1999 (1)

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[Crossref]

1998 (2)

S. Brasselet and J. Zyss, “Multipolar molecules and multipolar fields: probing and controlling the tensorial nature of nonlinear molecular media,” J. Opt. Soc. Am. B 15, 257–288 (1998).
[Crossref]

B. del Rey, U. Keller, T. Torres, G. Rojo, F. Agulló-López, S. Nonell, C. Martí, S. Brasselet, I. Ledoux, and J. Zyss, “Synthesis and nonlinear optical, photophysical, and electrochemical properties of subphthalocyanines,” J. Am. Chem. Soc. 120, 12808–12811 (1998).
[Crossref]

1997 (2)

I. R. Whittall, M. G. Humphrey, S. Houbrechts, J. Maes, A. Persoons, S. Schmid, and D. C. Hockless, “Organometallic complexes for nonlinear optics. 14. syntheses and second-order nonlinear optical properties of ruthenium, nickel and gold σ-acetylides of 1,3,5-triethynylbenzene: X-ray crystal structures of 1-(HC ≡ C)-3,5-C6H3 (trans-C ≡ CRuCl(dppm)2)2 and 1,3,5-C6H3(C ≡ CAu(PPh3))3,” J. Orgonomet. Chem. 544, 277–283 (1997).
[Crossref]

R. Wortmann, C. Glania, P. Kramer, R. Matschiner, J. J. Wolff, S. Kraft, B. Treptow, E. Barbu, D. Langle, and G. Gorlitz, “Nondipolar structures with threefold symmetry for nonlinear optics,” Chem. -Eur. J. 3, 1765–1773 (1997).
[Crossref]

1995 (1)

S. Stadler, F. Feiner, C. Brauchle, S. Brandl, and R. Gompper, “Determination of the first hyperpolarizability of four octupolar molecules and their dipolar subunits via hyper-rayleigh scattering in solution,” Chem. Phys. Lett. 245, 292–296 (1995).
[Crossref]

1994 (1)

W. Hübner, K. H. Bennemann, and K. Böhmer, “Theory for the nonlinear optical response of transition metals: Polarization dependence as a fingerprint of the electronic structure at surfaces and interfaces,” Phys. Rev. B 50, 17597 (1994).
[Crossref]

1993 (1)

G. Petrocelli, S. Martellucci, and R. Francini, “Wavelength dependence of second-harmonic generation at the copper surface,” Appl. Phys. A 56, 263–266 (1993).
[Crossref]

1992 (1)

1990 (1)

I. Ledoux, J. Zyss, J. Siegel, J. Brienne, and J. Lehnb, “Second-harmonic generation from non-dipolar non-centrosymmetric aromatic charge-transfer molecules,” Chem. Phys. Lett. 172, 440–444 (1990).
[Crossref]

1981 (1)

G. S. Agarwal and S. S. Jha, “Theory of second harmonic generation at a metal surface with surface plasmon excitation,” Solid State Commun. 41, 499–501 (1981).
[Crossref]

1977 (1)

J. L. Oudar and D. S. Chemla, “Hyperpolarizabilities of the nitroanilines and their relations to the excited state dipole moment,” J. Chem. Phys. 66, 2664–2668 (1977).
[Crossref]

1970 (1)

J. Jerphagnon, “Invariants of the third-rank cartesian tensor: Optical nonlinear susceptibilities,” Phys. Rev. B 2, 1091–1098 (1970).
[Crossref]

1968 (1)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. Lett. 174, 813–822 (1968).

1959 (1)

B. Richards, E. Wolf, and D. Gabor, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” P. Roy. Soc. London A Mat. 253, 358–379 (1959).

Adam, P.-M.

C. Hubert, L. Billot, P.-M. Adam, R. Bachelot, P. Royer, J. Grand, D. Gindre, K. D. Dorkenoo, and A. Fort, “Role of surface plasmon in second harmonic generation from gold nanorods,” Appl. Phys. Lett. 90, 181105 (2007).
[Crossref]

J. Grand, S. Kostcheev, J.-L. Bijeon, M. Lamy de la Chapelle, P.-M. Adam, A. Rumyantseva, G. Lerondel, and P. Royer, “Optimization of sers-active substrates for near-field raman spectroscopy,” Synth. Met. 139, 621–624 (2003).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and S. S. Jha, “Theory of second harmonic generation at a metal surface with surface plasmon excitation,” Solid State Commun. 41, 499–501 (1981).
[Crossref]

Agulló-López, F.

B. del Rey, U. Keller, T. Torres, G. Rojo, F. Agulló-López, S. Nonell, C. Martí, S. Brasselet, I. Ledoux, and J. Zyss, “Synthesis and nonlinear optical, photophysical, and electrochemical properties of subphthalocyanines,” J. Am. Chem. Soc. 120, 12808–12811 (1998).
[Crossref]

Antoine, I. R.

G. Bachelier, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” Phys. Rev. 25, 955–960 (2008).

J. Nappa, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Wavelength dependence of the retardation effects in silver nanoparticles followed by polarization resolved hyper rayleigh scattering,” Chem. Phys. Lett. 415, 246–250 (2005).
[Crossref]

Aouani, H.

S. D. Gennaro, M. Rahmani, V. Giannini, H. Aouani, T. P. Sidiropoulos, M. Navarro-Cía, S. A. Maier, and R. F. Oulton, “The interplay of symmetry and scattering phase in second harmonic generation from gold nanoantennas,” Nano Lett. 16, 5278–5285 (2016).
[Crossref] [PubMed]

Bachelier, G.

G. Bachelier, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” Phys. Rev. 25, 955–960 (2008).

Bachelot, R.

C. Hubert, L. Billot, P.-M. Adam, R. Bachelot, P. Royer, J. Grand, D. Gindre, K. D. Dorkenoo, and A. Fort, “Role of surface plasmon in second harmonic generation from gold nanorods,” Appl. Phys. Lett. 90, 181105 (2007).
[Crossref]

Bai, B.

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
[Crossref] [PubMed]

Barbu, E.

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Figures (9)

Fig. 1
Fig. 1 (a) Scanning Electron Microscopy (SEM) images of nanostars. The metallic nanoparticles have been lithographed far enough from each other with the purpose of studying them independently. (b) Geometry used for positioning the nanoparticles in the sample plane (X,Y) for polarization resolved and efficiency measurements. (c) Schematic of the polarization resolved SHG experiment. (d) SHG scanning image of an array of nanostars. The scale represents the sum of the SHG signals over 32 incident polarization angles, in counts/100 μs, pixel size: 60 nm. (e) Emission spectrum from a single nanostar. Integration time: 1s.
Fig. 2
Fig. 2 Histograms of the off-resonant χ ( 2 ) for nanocylinders, nanotriangles and nanostars, obtained by comparison with a bulk KTP crystal measurement (see Appendix B). 45 particles were measured for each shape.
Fig. 3
Fig. 3 The radiation of three SH induced dipoles through the collection objective of microscope giving rise to the three SHG field images.
Fig. 4
Fig. 4 SHG signals as a function of the polarization angle along the X and Y axes (sX in blue and sY in red) with various values of c corresponding to the distance h from the center of star to the position where the SH induced dipole sources located. Two lobes polar patterns are a signature of non-overlapping signals whereas four lobes ones denote fully overlapping signals, typical of a threefold symmetry [37].
Fig. 5
Fig. 5 (a) Schematic of the two fitting models. Model 1: Three dipoles of strength β located at a distance h from the center and separated by 2 π 3. Model 2: Three dipoles of different strengths β 1 , β 2 and β3 and arbitrary angles ν1 and ν2. Fitting results obtained on the data of two typical nanostars (b), (c) and two typical triangles (d), (e). Particle (b) and (d) have an almost perfect threefold symmetry while (c) and (e) are deformed. Fitting parameters from the models are given in Appendix, table 2.
Fig. 6
Fig. 6 Histogram of intensities collected from a population of 45 individual nanostars (top), nanotriangles (middle), and nanocylinders (bottom), with an incident averaged power of 0.4 mW at the focal spot of the objective. The obtained average and standard deviation values are: (140 090 ± 52 200) counts/s (nanostars), (84 800 ± 27 800) counts/s (nanotriangles), (1 520 ± 1 040) counts/s (nanocylinders).
Fig. 7
Fig. 7 (a) Histogram of the nonlinear coefficients χ ( 2 ) obtained by comparison with a bulk KTP crystal measurement (see text). (b) Histogram of χ ( 2 ) values corrected from the resonance to non-resonance factors obtained for the nanoparticles.
Fig. 8
Fig. 8 Schematic of the overlap of the point spread function (PSF) for two dipoles: (a) g 1 ( r ) g 2 ( r ) d s = 0, two PSF do not overlap. This case occurs when the distance between two dipoles is large. (b) g 1 ( r ) g 2 ( r ) d s = 0.5, two PSF overlap partially and (c) g 1 ( r ) g 2 ( r ) d s = 1, two PSF overlap fully. Case (c) occurs when two dipoles are located at the same position.
Fig. 9
Fig. 9 Evolution of the overlap parameter c as a function of the distance h of translation of the dipole from the center of the star. When h = 0, the 3 dipoles are on top of each other and then interfere constructively. As h increases, the overlap decreases and the dipoles start to interfere destructively.

Tables (3)

Tables Icon

Table 1 Relative weights of the different multipole orders for various values of the nanostar size h. The model depends on three identical dipoles oriented in a threefold symmetry structure and a variable distance from the nano-star center.

Tables Icon

Table 2 Parameters describing the susceptibilities of nanostars, nano triangle and nanocylinders obtained from experimental fits.

Tables Icon

Table 3 This table summarizes the fitting parameters that have been calculated according to the two models given in the main text. Moreover, the right panel provides multipolar weight for each nanoparticles. These fits have been carried systematically for all nanoparticles and we provide here four typical cases. Stars (triangles) 1 and 2 refer to a perfect and a deformed nanostars (nanotriangles) respectively. For the model 2, the multipolar distribution is dependent on the angle α of the incoming field polarization and we thus give the multipolar weight for the stars for polarization that maximize the dipolar weight (noted dip) and the quadrupolar one (noted quad). The multipolar distribution of the triangle is quite robust to shape deformation and does not vary much with the polarization. For any case, β 0 = 0 and γ = 0.

Equations (41)

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μ j = ε 0 β ( E ( ω ) e j ) ( E ω e j ) e j
c = 4 π 0 1 sin  ( 3 M a h 1 v 2 ) 3 M a h d v
Ψ m J M ( r , θ , φ ) = 1 J ( J + 1 ) j J ( k r ) L Y J M ( θ , φ )
Ψ e J M ( r , θ , φ ) = i k Ψ m J M ( r , θ , φ )
J i J M ( c ) = J ( c , r ) Ψ i J M d V
J J ( c ) = i M ( 1 ) M J i J M ( c ) J i J M ( c ) J i M ( 1 ) M J i J M ( c ) J i J M ( c )
I ( 2 ω ) = C . ( V exc ) 2 . ( χ ( 2 ) ) 2 . ( I ( ω ) ) 2
I KTP ( 2 ω ) = C . ( V exc KTP ) 2 . ( χ KTP ( 2 ) ) 2 . ( I KTP ( ω ) ) 2
I particle ( 2 ω ) = C . ( V exc particle ) 2 . ( χ particle ( 2 ) ) 2 . ( I particle ( ω ) ) 2
χ particle ( 2 ) = χ Y Y Y , KTP ( 2 ) . I particle ( 2 ω ) I KTP ( 2 ω ) . I particle ( ω ) I KTP ( ω ) . V exc KTP V exc particle
k = 2 k ω + k 2 ω = 2 2 π λ ( n ω + n 2 ω ) and k l coherent 2 = π 2
E ( r ) = ω 2 ε 0 c 2 G ( r , r n ) μ
G P S F ( r ) = K ( A 0 ( r ) 0 0 0 A 0 ( r ) 0 0 0 0 )
A 0 ( r ) = 2 M θ m k r J 1 ( k r θ m M )
E ( r M r n ) = ω 2 ε 0 c 2 [   G P S F ( r ) μ ] * δ ( r M r n )
μ j ( 2 ω ) ( α ) = ε 0 β j . [ E j ( ω ) ( α ) ] 2
E j ( 2 ω ) ( r ) = ω 2 ε 0 2 c 2 [ G ( r ) μ j ( 2 ω ) ] * δ ( r M r n )
E j ( 2 ω ) ( r , α ) = K ω 2 ε 0 2 c 2 ( A 0 ( r ) 0 0 0 A 0 ( r ) 0 0 0 0 ) ( μ j X ( 2 ω ) ( α ) μ j Y ( 2 ω ) ( α ) μ j Z ( 2 ω ) ( α ) )
E j p ( 2 ω ) ( r , α ) = K ω 2 ε 0 2 c 2 A 0 ( r ) μ j p ( 2 ω ) ( α )
s p = S dect I p d s = S dect | j E j p ( 2 ω ) ( r , α ) | 2 d s
I 1 p ( r ) = g 1 p 2 ( r ) s 1 p and g 1 p 2 ( r ) d s = 1
I 1 p ( r ) = ( E 1 p ( 2 ω ) ( r , α ) ) 2 = K 2 ω 4 ε 0 4 c 4 A 0 2 ( r ) ( μ 1 p ( 2 ω ) ( α ) ) 2
s 1 p = S dect I 1 p ( r ) d s = ( E 1 p ( 2 ω ) ( r , α ) ) 2 d s = K 2 ω 4 ε 0 4 c 4 ( μ 1 p ( 2 ω ) ( α ) ) 2 A 0 2 ( r ) d s
g 1 p ( r ) = A 0 ( r ) A 0 2 ( r ) d s g 1 p ( r ) = A 0 ( r ) μ 1 p ( 2 ω ) ( α ) s 1 p and g 1 p ( r ) = E 1 p ( 2 ω ) s 1 p
s p = K 2 ω 4 ε 0 4 c 4 ( μ 1 p ( 2 ω ) ( α ) A 01 ( r ) + μ 2 p ( 2 ω ) ( α ) A 02 ( r ) ) 2 d s
s p = ( s 1 p g 1 p 2 ( r ) + s 2 p g 2 p 2 ( r ) + 2 s 1 p s 2 p g 1 p ( r ) g 2 p ( r ) ) d s = s 1 p + s 2 p + 2 s 1 p s 2 p g 1 p ( r ) g 2 p ( r ) d s = ( 1 g 1 p ( r ) g 2 p ( r ) d s ) ( s 1 p + s 2 p ) + ( g 1 p ( r ) g 2 p ( r ) d s ) ( s 1 p + s 2 p ) 2
s p = ( 1 g 1 p ( r ) g 2 p ( r ) d s ) s n o v p + ( g 1 p ( r ) g 2 p ( r ) d s ) s f o v p
c = g 1 p ( r ) g 2 p ( r ) d s = A 01 ( r ) A 02 ( r ) d s A 01 2 ( r ) d s A 02 2 ( r ) d s
s p = ( j μ j p ( 2 ω ) ( α ) A 0 j ( r ) ) 2 d s = j s j p + 2 i , j s i p s j p g i p ( r ) g j p ( r ) d s
s p = ( 1 g i p ( r ) g j p ( r ) d s ) j s j p + ( g i p ( r ) g j p d s ) ( j s j p ) 2
s p = ( 1 g i p ( r ) g j p ( r ) d s ) s i n p + ( g i p ( r ) g j p ( r ) d s ) s c o p
c i j = g i p ( r ) g j p ( r ) d s = A 0 i ( r ) A 0 j ( r ) d s A 0 i 2 ( r ) d s A 0 j 2 ( r ) d s
c = A 0 i ( r ) A 0 j ( r ) d s A 0 i 2 ( r ) d s A 0 j 2 ( r ) d s = J 1 ( a r ) r J 1 ( a ( r r 0 ) ) r r 0 d s ( J 1 ( a r ) r ) 2 d s
c = J 1 ( a x 2 + y 2 ) x 2 + y 2 J 1 ( a ( x x 0 ) 2 + y 2 ) ( x x 0 ) 2 + y 2 d s ( J 1 ( a x 2 + y 2 ) x 2 + y 2 ) 2 d s
F ( J 1 ( a x 2 + y 2 ) x 2 + y 2 ) = 2 π a circ ( 2 π a ξ x 2 + ξ y 2 )
F ( J 1 ( a ( x x 0 ) 2 + y 2 ) ( x x 0 ) 2 + y 2 ) = e 2 i π x 0 ξ x 2 π a circ ( 2 π a ξ x 2 + ξ y 2 )
c = h ( x 0 ) h ( 0 ) with h ( x 0 ) = J 1 ( a x 2 + y 2 ) x 2 + y 2 J 1 ( a ( x x 0 ) 2 + y 2 ) ( x x 0 ) 2 + y 2 d x d y
h ( x 0 ) = e 2 i π x 0 ξ x ( 2 π a ) 2 circ 2 ( 2 π a ξ x 2 + ξ y 2 ) d ξ x d ξ y = e 2 i π x 0 a 2 π u circ 2 ( u 2 + v 2 ) d u d v
h ( x 0 ) = 1 1 1 1 e i a x 0 u circ u 2 + v 2 d u d v = 4 0 1 sin  ( a x 0 1 v 2 ) a x 0 d v
c = 4 π 0 1 sin  ( a x 0 1 v 2 ) a x 0 d v = 4 π 0 1 sin  ( a M d 1 v 2 ) a M d d v
c = 4 π 0 1 sin  ( 3 M a h 1 v 2 ) 3 M a h d v

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