Abstract

We derive the leading-order correction to the proximity force approximation (PFA) result for the electromagnetic Casimir interaction in the plane–sphere geometry by developing the scattering approach in the plane-wave basis. Expressing the Casimir energy as a sum over round trips between plane and sphere, we find two distinct contributions to the correction. The first one results from the variation of the Mie reflection operator, calculated within the geometric optical Wentzel–Kramers–Brillouin (WKB) approximation, over the narrow Fourier interval associated with specular reflection at the vicinity of the point of closest approach on the spherical surface. The second contribution, accounting for roughly 90% of the total correction, results from the modification of the geometric optical WKB Mie scattering amplitude due to diffraction. Our derivation recovers the known leading-order correction to the PFA and shows that all contributing scattering channels are of a local nature.

© 2019 Optical Society of America

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References

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2018 (3)

B. Spreng, M. Hartmann, V. Henning, P. A. Maia Neto, and G.-L. Ingold, “Proximity force approximation and specular reflection: application of the WKB limit of Mie scattering to the Casimir effect,” Phys. Rev. A 97, 062504 (2018).
[Crossref]

M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, “Advancing numerics for the Casimir effect to experimentally relevant aspect ratios,” Phys. Scr. 93, 114003 (2018).
[Crossref]

J. L. Garrett, D. A. T. Somers, and J. N. Munday, “Measurement of the Casimir force between two spheres,” Phys. Rev. Lett. 120, 040401 (2018).
[Crossref]

2017 (2)

M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, “Plasma versus Drude modeling of the Casimir force: beyond the proximity force approximation,” Phys. Rev. Lett. 119, 043901 (2017).
[Crossref]

G. Bimonte, “Classical Casimir interaction of perfectly conducting sphere and plate,” Phys. Rev. D 95, 065004 (2017).
[Crossref]

2016 (2)

M. Sedighi, V. B. Svetovoy, and G. Palasantzas, “Casimir force measurements from silicon carbide surfaces,” Phys. Rev. B 93, 085434 (2016).
[Crossref]

G. Bimonte, D. López, and R. S. Decca, “Isoelectronic determination of the thermal Casimir force,” Phys. Rev. B 93, 184434 (2016).
[Crossref]

2015 (5)

D. S. Ether, L. B. Pires, S. Umrath, D. Martinez, Y. Ayala, B. Pontes, G. R. de S. Araújo, S. Frases, G.-L. Ingold, F. S. S. Rosa, N. B. Viana, H. M. Nussenzveig, and P. A. Maia Neto, “Probing the Casimir force with optical tweezers,” Europhys. Lett. 112, 44001 (2015).
[Crossref]

C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Derivative expansion for the electromagnetic Casimir free energy at high temperatures,” Phys. Rev. D 92, 125007 (2015).
[Crossref]

K. A. Milton, R. Guérout, G.-L. Ingold, A. Lambrecht, and S. Reynaud, “Negative Casimir entropies in nanoparticle interactions,” J. Phys. Condens. Matter 27, 214003 (2015).
[Crossref]

G.-L. Ingold, S. Umrath, M. Hartmann, R. Guérout, A. Lambrecht, S. Reynaud, and K. A. Milton, “Geometric origin of negative Casimir entropies: a scattering-channel analysis,” Phys. Rev. E 91, 033203 (2015).
[Crossref]

S. Umrath, M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, “Disentangling geometric and dissipative origins of negative Casimir entropies,” Phys. Rev. E 92, 042125 (2015).
[Crossref]

2014 (2)

C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Derivative-expansion approach to the interaction between close surfaces,” Phys. Rev. A 89, 062120 (2014).
[Crossref]

M. Elzbieciak-Wodka, M. N. Popescu, F. J. M. Ruiz-Cabello, G. Trefalt, P. Maroni, and M. Borkovec, “Measurements of dispersion forces between colloidal latex particles with the atomic force microscope and comparison with Lifshitz theory,” J. Chem. Phys. 140, 104906 (2014).
[Crossref]

2013 (2)

A. A. Banishev, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Demonstration of the Casimir force between ferromagnetic surfaces of a Ni-coated sphere and a Ni-coated plate,” Phys. Rev. Lett. 110, 137401 (2013).
[Crossref]

L. P. Teo, “Material dependence of Casimir interaction between a sphere and a plate: first analytic correction beyond proximity force approximation,” Phys. Rev. D 88, 045019 (2013).
[Crossref]

2012 (5)

G. Bimonte and T. Emig, “Exact results for classical Casimir interactions: Dirichlet and Drude model in the sphere-sphere and sphere-plane geometry,” Phys. Rev. Lett. 109, 160403 (2012).
[Crossref]

G. Bimonte, T. Emig, and M. Kardar, “Material dependence of Casimir forces: gradient expansion beyond proximity,” Appl. Phys. Lett. 100, 074110 (2012).
[Crossref]

C.-C. Chang, A. A. Banishev, R. Castillo-Garza, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Gradient of the Casimir force between Au surfaces of a sphere and a plate measured using an atomic force microscope in a frequency-shift technique,” Phys. Rev. B 85, 165443 (2012).
[Crossref]

D. Garcia-Sanchez, K. Y. Fong, H. Bhaskaran, S. Lamoreaux, and H. X. Tang, “Casimir force and in situ surface potential measurements on nanomembranes,” Phys. Rev. Lett. 109, 027202 (2012).
[Crossref]

G. Bimonte, T. Emig, R. L. Jaffe, and M. Kardar, “Casimir forces beyond the proximity approximation,” Europhys. Lett. 97, 50001(2012).
[Crossref]

2011 (6)

C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Proximity force approximation for the Casimir energy as a derivative expansion,” Phys. Rev. D 84, 105031 (2011).
[Crossref]

L. P. Teo, M. Bordag, and V. Nikolaev, “Corrections beyond the proximity force approximation,” Phys. Rev. D 84, 125037 (2011).
[Crossref]

R. Decca, V. Aksyuk, and D. López, “Casimir force in micro and nano electro mechanical systems,” Lect. Notes Phys. 834, 287–309 (2011).
[Crossref]

S. K. Lamoreaux, “Progress in experimental measurements of the surface-surface Casimir force: electrostatic calibrations and limitations to accuracy,” Lect. Notes Phys. 834, 219–248 (2011).
[Crossref]

A. O. Sushkov, W. J. Kim, D. A. R. Dalvit, and S. K. Lamoreaux, “Observation of the thermal Casimir force,” Nat. Phys. 7, 230–233 (2011).
[Crossref]

G. Torricelli, I. Pirozhenko, S. Thornton, A. Lambrecht, and C. Binns, “Casimir force between a metal and a semimetal,” Europhys. Lett. 93, 51001 (2011).
[Crossref]

2010 (4)

A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Thermal Casimir effect in the plane-sphere geometry,” Phys. Rev. Lett. 104, 040403 (2010).
[Crossref]

A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Thermal Casimir effect for Drude metals in the plane-sphere geometry,” Phys. Rev. A 82, 012511 (2010).
[Crossref]

R. Zandi, T. Emig, and U. Mohideen, “Quantum and thermal Casimir interaction between a sphere and a plate: comparison of Drude and plasma models,” Phys. Rev. B 81, 195423 (2010).
[Crossref]

M. Bordag and I. Pirozhenko, “Vacuum energy between a sphere and a plane at finite temperature,” Phys. Rev. D 81, 085023 (2010).
[Crossref]

2009 (2)

A. Canaguier-Durand, P. A. Maia Neto, I. Cavero-Pelaez, A. Lambrecht, and S. Reynaud, “Casimir interaction between plane and spherical metallic surfaces,” Phys. Rev. Lett. 102, 230404 (2009).
[Crossref]

G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “The Casimir force between real materials: experiment and theory,” Rev. Mod. Phys. 81, 1827–1885 (2009).
[Crossref]

2008 (3)

P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Casimir energy between a plane and a sphere in electromagnetic vacuum,” Phys. Rev. A 78, 012115 (2008).
[Crossref]

T. Emig, “Fluctuation-induced quantum interactions between compact objects and a plane mirror,” J. Stat. Mech. 2008P04007 (2008).
[Crossref]

M. Bordag and V. Nikolaev, “Casimir force for a sphere in front of a plane beyond proximity force approximation,” J. Phys. A 41, 164002 (2008).
[Crossref]

2007 (1)

T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, “Casimir forces between arbitrary compact objects,” Phys. Rev. Lett. 99, 170403 (2007).
[Crossref]

2006 (1)

A. Lambrecht, P. A. Maia Neto, and S. Reynaud, “The Casimir effect within scattering theory,” New J. Phys. 8, 243 (2006).
[Crossref]

2005 (1)

A. Scardicchio and R. L. Jaffe, “Casimir effects: an optical approach I. Foundations and examples,” Nucl. Phys. B 704, 552–582 (2005).
[Crossref]

2004 (1)

R. L. Jaffe and A. Scardicchio, “Casimir effect and geometric optics,” Phys. Rev. Lett. 92, 070402 (2004).
[Crossref]

2002 (1)

B. C. Berndt and B. P. Yeap, “Explicit evaluations and reciprocity theorems for finite trigonometric sums,” Adv. Appl. Math. 29, 358–385 (2002).
[Crossref]

1998 (1)

M. Schaden and L. Spruch, “Infinity-free semiclassical evaluation of Casimir effects,” Phys. Rev. A 58, 935–953 (1998).
[Crossref]

1969 (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[Crossref]

1948 (1)

H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proc. K. Ned. Akad. Wet. 51, 793–795 (1948).

1934 (1)

B. Derjaguin, “Untersuchungen über die Reibung und Adhäsion, IV—Theorie des Anhaftens kleiner Teilchen,” Kolloid-Zs. 69, 155–164 (1934).
[Crossref]

Aksyuk, V.

R. Decca, V. Aksyuk, and D. López, “Casimir force in micro and nano electro mechanical systems,” Lect. Notes Phys. 834, 287–309 (2011).
[Crossref]

Ayala, Y.

D. S. Ether, L. B. Pires, S. Umrath, D. Martinez, Y. Ayala, B. Pontes, G. R. de S. Araújo, S. Frases, G.-L. Ingold, F. S. S. Rosa, N. B. Viana, H. M. Nussenzveig, and P. A. Maia Neto, “Probing the Casimir force with optical tweezers,” Europhys. Lett. 112, 44001 (2015).
[Crossref]

Banishev, A. A.

A. A. Banishev, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Demonstration of the Casimir force between ferromagnetic surfaces of a Ni-coated sphere and a Ni-coated plate,” Phys. Rev. Lett. 110, 137401 (2013).
[Crossref]

C.-C. Chang, A. A. Banishev, R. Castillo-Garza, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Gradient of the Casimir force between Au surfaces of a sphere and a plate measured using an atomic force microscope in a frequency-shift technique,” Phys. Rev. B 85, 165443 (2012).
[Crossref]

Berndt, B. C.

B. C. Berndt and B. P. Yeap, “Explicit evaluations and reciprocity theorems for finite trigonometric sums,” Adv. Appl. Math. 29, 358–385 (2002).
[Crossref]

Bhaskaran, H.

D. Garcia-Sanchez, K. Y. Fong, H. Bhaskaran, S. Lamoreaux, and H. X. Tang, “Casimir force and in situ surface potential measurements on nanomembranes,” Phys. Rev. Lett. 109, 027202 (2012).
[Crossref]

Bimonte, G.

G. Bimonte, “Classical Casimir interaction of perfectly conducting sphere and plate,” Phys. Rev. D 95, 065004 (2017).
[Crossref]

G. Bimonte, D. López, and R. S. Decca, “Isoelectronic determination of the thermal Casimir force,” Phys. Rev. B 93, 184434 (2016).
[Crossref]

G. Bimonte and T. Emig, “Exact results for classical Casimir interactions: Dirichlet and Drude model in the sphere-sphere and sphere-plane geometry,” Phys. Rev. Lett. 109, 160403 (2012).
[Crossref]

G. Bimonte, T. Emig, and M. Kardar, “Material dependence of Casimir forces: gradient expansion beyond proximity,” Appl. Phys. Lett. 100, 074110 (2012).
[Crossref]

G. Bimonte, T. Emig, R. L. Jaffe, and M. Kardar, “Casimir forces beyond the proximity approximation,” Europhys. Lett. 97, 50001(2012).
[Crossref]

Binns, C.

G. Torricelli, I. Pirozhenko, S. Thornton, A. Lambrecht, and C. Binns, “Casimir force between a metal and a semimetal,” Europhys. Lett. 93, 51001 (2011).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bordag, M.

L. P. Teo, M. Bordag, and V. Nikolaev, “Corrections beyond the proximity force approximation,” Phys. Rev. D 84, 125037 (2011).
[Crossref]

M. Bordag and I. Pirozhenko, “Vacuum energy between a sphere and a plane at finite temperature,” Phys. Rev. D 81, 085023 (2010).
[Crossref]

M. Bordag and V. Nikolaev, “Casimir force for a sphere in front of a plane beyond proximity force approximation,” J. Phys. A 41, 164002 (2008).
[Crossref]

M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir Effect (Oxford University, 2009).

Borkovec, M.

M. Elzbieciak-Wodka, M. N. Popescu, F. J. M. Ruiz-Cabello, G. Trefalt, P. Maroni, and M. Borkovec, “Measurements of dispersion forces between colloidal latex particles with the atomic force microscope and comparison with Lifshitz theory,” J. Chem. Phys. 140, 104906 (2014).
[Crossref]

Butt, H.-J.

H.-J. Butt and M. Kappl, Surface and Interfacial Forces (Wiley-VCH Verlag, 2010).

Canaguier-Durand, A.

A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Thermal Casimir effect in the plane-sphere geometry,” Phys. Rev. Lett. 104, 040403 (2010).
[Crossref]

A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Thermal Casimir effect for Drude metals in the plane-sphere geometry,” Phys. Rev. A 82, 012511 (2010).
[Crossref]

A. Canaguier-Durand, P. A. Maia Neto, I. Cavero-Pelaez, A. Lambrecht, and S. Reynaud, “Casimir interaction between plane and spherical metallic surfaces,” Phys. Rev. Lett. 102, 230404 (2009).
[Crossref]

Casimir, H. B. G.

H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proc. K. Ned. Akad. Wet. 51, 793–795 (1948).

Castillo-Garza, R.

C.-C. Chang, A. A. Banishev, R. Castillo-Garza, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Gradient of the Casimir force between Au surfaces of a sphere and a plate measured using an atomic force microscope in a frequency-shift technique,” Phys. Rev. B 85, 165443 (2012).
[Crossref]

Cavero-Pelaez, I.

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G. Bimonte, D. López, and R. S. Decca, “Isoelectronic determination of the thermal Casimir force,” Phys. Rev. B 93, 184434 (2016).
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G. Bimonte, T. Emig, R. L. Jaffe, and M. Kardar, “Casimir forces beyond the proximity approximation,” Europhys. Lett. 97, 50001(2012).
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D. Garcia-Sanchez, K. Y. Fong, H. Bhaskaran, S. Lamoreaux, and H. X. Tang, “Casimir force and in situ surface potential measurements on nanomembranes,” Phys. Rev. Lett. 109, 027202 (2012).
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C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Derivative expansion for the electromagnetic Casimir free energy at high temperatures,” Phys. Rev. D 92, 125007 (2015).
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C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Proximity force approximation for the Casimir energy as a derivative expansion,” Phys. Rev. D 84, 105031 (2011).
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G.-L. Ingold, S. Umrath, M. Hartmann, R. Guérout, A. Lambrecht, S. Reynaud, and K. A. Milton, “Geometric origin of negative Casimir entropies: a scattering-channel analysis,” Phys. Rev. E 91, 033203 (2015).
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B. Spreng, M. Hartmann, V. Henning, P. A. Maia Neto, and G.-L. Ingold, “Proximity force approximation and specular reflection: application of the WKB limit of Mie scattering to the Casimir effect,” Phys. Rev. A 97, 062504 (2018).
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G.-L. Ingold, S. Umrath, M. Hartmann, R. Guérout, A. Lambrecht, S. Reynaud, and K. A. Milton, “Geometric origin of negative Casimir entropies: a scattering-channel analysis,” Phys. Rev. E 91, 033203 (2015).
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M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, “Advancing numerics for the Casimir effect to experimentally relevant aspect ratios,” Phys. Scr. 93, 114003 (2018).
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B. Spreng, M. Hartmann, V. Henning, P. A. Maia Neto, and G.-L. Ingold, “Proximity force approximation and specular reflection: application of the WKB limit of Mie scattering to the Casimir effect,” Phys. Rev. A 97, 062504 (2018).
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M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, “Plasma versus Drude modeling of the Casimir force: beyond the proximity force approximation,” Phys. Rev. Lett. 119, 043901 (2017).
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D. S. Ether, L. B. Pires, S. Umrath, D. Martinez, Y. Ayala, B. Pontes, G. R. de S. Araújo, S. Frases, G.-L. Ingold, F. S. S. Rosa, N. B. Viana, H. M. Nussenzveig, and P. A. Maia Neto, “Probing the Casimir force with optical tweezers,” Europhys. Lett. 112, 44001 (2015).
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G.-L. Ingold, S. Umrath, M. Hartmann, R. Guérout, A. Lambrecht, S. Reynaud, and K. A. Milton, “Geometric origin of negative Casimir entropies: a scattering-channel analysis,” Phys. Rev. E 91, 033203 (2015).
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G. Bimonte, T. Emig, R. L. Jaffe, and M. Kardar, “Casimir forces beyond the proximity approximation,” Europhys. Lett. 97, 50001(2012).
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G. Bimonte, T. Emig, R. L. Jaffe, and M. Kardar, “Casimir forces beyond the proximity approximation,” Europhys. Lett. 97, 50001(2012).
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G. Bimonte, T. Emig, and M. Kardar, “Material dependence of Casimir forces: gradient expansion beyond proximity,” Appl. Phys. Lett. 100, 074110 (2012).
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T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, “Casimir forces between arbitrary compact objects,” Phys. Rev. Lett. 99, 170403 (2007).
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A. O. Sushkov, W. J. Kim, D. A. R. Dalvit, and S. K. Lamoreaux, “Observation of the thermal Casimir force,” Nat. Phys. 7, 230–233 (2011).
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A. A. Banishev, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Demonstration of the Casimir force between ferromagnetic surfaces of a Ni-coated sphere and a Ni-coated plate,” Phys. Rev. Lett. 110, 137401 (2013).
[Crossref]

C.-C. Chang, A. A. Banishev, R. Castillo-Garza, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Gradient of the Casimir force between Au surfaces of a sphere and a plate measured using an atomic force microscope in a frequency-shift technique,” Phys. Rev. B 85, 165443 (2012).
[Crossref]

G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “The Casimir force between real materials: experiment and theory,” Rev. Mod. Phys. 81, 1827–1885 (2009).
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M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir Effect (Oxford University, 2009).

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K. A. Milton, R. Guérout, G.-L. Ingold, A. Lambrecht, and S. Reynaud, “Negative Casimir entropies in nanoparticle interactions,” J. Phys. Condens. Matter 27, 214003 (2015).
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G.-L. Ingold, S. Umrath, M. Hartmann, R. Guérout, A. Lambrecht, S. Reynaud, and K. A. Milton, “Geometric origin of negative Casimir entropies: a scattering-channel analysis,” Phys. Rev. E 91, 033203 (2015).
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G. Torricelli, I. Pirozhenko, S. Thornton, A. Lambrecht, and C. Binns, “Casimir force between a metal and a semimetal,” Europhys. Lett. 93, 51001 (2011).
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A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Thermal Casimir effect for Drude metals in the plane-sphere geometry,” Phys. Rev. A 82, 012511 (2010).
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A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Thermal Casimir effect in the plane-sphere geometry,” Phys. Rev. Lett. 104, 040403 (2010).
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A. Canaguier-Durand, P. A. Maia Neto, I. Cavero-Pelaez, A. Lambrecht, and S. Reynaud, “Casimir interaction between plane and spherical metallic surfaces,” Phys. Rev. Lett. 102, 230404 (2009).
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P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Casimir energy between a plane and a sphere in electromagnetic vacuum,” Phys. Rev. A 78, 012115 (2008).
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A. Lambrecht, P. A. Maia Neto, and S. Reynaud, “The Casimir effect within scattering theory,” New J. Phys. 8, 243 (2006).
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D. Garcia-Sanchez, K. Y. Fong, H. Bhaskaran, S. Lamoreaux, and H. X. Tang, “Casimir force and in situ surface potential measurements on nanomembranes,” Phys. Rev. Lett. 109, 027202 (2012).
[Crossref]

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A. O. Sushkov, W. J. Kim, D. A. R. Dalvit, and S. K. Lamoreaux, “Observation of the thermal Casimir force,” Nat. Phys. 7, 230–233 (2011).
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S. K. Lamoreaux, “Progress in experimental measurements of the surface-surface Casimir force: electrostatic calibrations and limitations to accuracy,” Lect. Notes Phys. 834, 219–248 (2011).
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C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Derivative expansion for the electromagnetic Casimir free energy at high temperatures,” Phys. Rev. D 92, 125007 (2015).
[Crossref]

C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Derivative-expansion approach to the interaction between close surfaces,” Phys. Rev. A 89, 062120 (2014).
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C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Proximity force approximation for the Casimir energy as a derivative expansion,” Phys. Rev. D 84, 105031 (2011).
[Crossref]

López, D.

G. Bimonte, D. López, and R. S. Decca, “Isoelectronic determination of the thermal Casimir force,” Phys. Rev. B 93, 184434 (2016).
[Crossref]

R. Decca, V. Aksyuk, and D. López, “Casimir force in micro and nano electro mechanical systems,” Lect. Notes Phys. 834, 287–309 (2011).
[Crossref]

Maia Neto, P. A.

B. Spreng, M. Hartmann, V. Henning, P. A. Maia Neto, and G.-L. Ingold, “Proximity force approximation and specular reflection: application of the WKB limit of Mie scattering to the Casimir effect,” Phys. Rev. A 97, 062504 (2018).
[Crossref]

M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, “Advancing numerics for the Casimir effect to experimentally relevant aspect ratios,” Phys. Scr. 93, 114003 (2018).
[Crossref]

M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, “Plasma versus Drude modeling of the Casimir force: beyond the proximity force approximation,” Phys. Rev. Lett. 119, 043901 (2017).
[Crossref]

D. S. Ether, L. B. Pires, S. Umrath, D. Martinez, Y. Ayala, B. Pontes, G. R. de S. Araújo, S. Frases, G.-L. Ingold, F. S. S. Rosa, N. B. Viana, H. M. Nussenzveig, and P. A. Maia Neto, “Probing the Casimir force with optical tweezers,” Europhys. Lett. 112, 44001 (2015).
[Crossref]

S. Umrath, M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, “Disentangling geometric and dissipative origins of negative Casimir entropies,” Phys. Rev. E 92, 042125 (2015).
[Crossref]

A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Thermal Casimir effect in the plane-sphere geometry,” Phys. Rev. Lett. 104, 040403 (2010).
[Crossref]

A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Thermal Casimir effect for Drude metals in the plane-sphere geometry,” Phys. Rev. A 82, 012511 (2010).
[Crossref]

A. Canaguier-Durand, P. A. Maia Neto, I. Cavero-Pelaez, A. Lambrecht, and S. Reynaud, “Casimir interaction between plane and spherical metallic surfaces,” Phys. Rev. Lett. 102, 230404 (2009).
[Crossref]

P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Casimir energy between a plane and a sphere in electromagnetic vacuum,” Phys. Rev. A 78, 012115 (2008).
[Crossref]

A. Lambrecht, P. A. Maia Neto, and S. Reynaud, “The Casimir effect within scattering theory,” New J. Phys. 8, 243 (2006).
[Crossref]

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M. Elzbieciak-Wodka, M. N. Popescu, F. J. M. Ruiz-Cabello, G. Trefalt, P. Maroni, and M. Borkovec, “Measurements of dispersion forces between colloidal latex particles with the atomic force microscope and comparison with Lifshitz theory,” J. Chem. Phys. 140, 104906 (2014).
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D. S. Ether, L. B. Pires, S. Umrath, D. Martinez, Y. Ayala, B. Pontes, G. R. de S. Araújo, S. Frases, G.-L. Ingold, F. S. S. Rosa, N. B. Viana, H. M. Nussenzveig, and P. A. Maia Neto, “Probing the Casimir force with optical tweezers,” Europhys. Lett. 112, 44001 (2015).
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C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Derivative expansion for the electromagnetic Casimir free energy at high temperatures,” Phys. Rev. D 92, 125007 (2015).
[Crossref]

C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Derivative-expansion approach to the interaction between close surfaces,” Phys. Rev. A 89, 062120 (2014).
[Crossref]

C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Proximity force approximation for the Casimir energy as a derivative expansion,” Phys. Rev. D 84, 105031 (2011).
[Crossref]

Milton, K. A.

G.-L. Ingold, S. Umrath, M. Hartmann, R. Guérout, A. Lambrecht, S. Reynaud, and K. A. Milton, “Geometric origin of negative Casimir entropies: a scattering-channel analysis,” Phys. Rev. E 91, 033203 (2015).
[Crossref]

K. A. Milton, R. Guérout, G.-L. Ingold, A. Lambrecht, and S. Reynaud, “Negative Casimir entropies in nanoparticle interactions,” J. Phys. Condens. Matter 27, 214003 (2015).
[Crossref]

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A. A. Banishev, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Demonstration of the Casimir force between ferromagnetic surfaces of a Ni-coated sphere and a Ni-coated plate,” Phys. Rev. Lett. 110, 137401 (2013).
[Crossref]

C.-C. Chang, A. A. Banishev, R. Castillo-Garza, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Gradient of the Casimir force between Au surfaces of a sphere and a plate measured using an atomic force microscope in a frequency-shift technique,” Phys. Rev. B 85, 165443 (2012).
[Crossref]

R. Zandi, T. Emig, and U. Mohideen, “Quantum and thermal Casimir interaction between a sphere and a plate: comparison of Drude and plasma models,” Phys. Rev. B 81, 195423 (2010).
[Crossref]

G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “The Casimir force between real materials: experiment and theory,” Rev. Mod. Phys. 81, 1827–1885 (2009).
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M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir Effect (Oxford University, 2009).

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A. A. Banishev, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Demonstration of the Casimir force between ferromagnetic surfaces of a Ni-coated sphere and a Ni-coated plate,” Phys. Rev. Lett. 110, 137401 (2013).
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C.-C. Chang, A. A. Banishev, R. Castillo-Garza, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Gradient of the Casimir force between Au surfaces of a sphere and a plate measured using an atomic force microscope in a frequency-shift technique,” Phys. Rev. B 85, 165443 (2012).
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G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “The Casimir force between real materials: experiment and theory,” Rev. Mod. Phys. 81, 1827–1885 (2009).
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M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir Effect (Oxford University, 2009).

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G. Torricelli, I. Pirozhenko, S. Thornton, A. Lambrecht, and C. Binns, “Casimir force between a metal and a semimetal,” Europhys. Lett. 93, 51001 (2011).
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D. S. Ether, L. B. Pires, S. Umrath, D. Martinez, Y. Ayala, B. Pontes, G. R. de S. Araújo, S. Frases, G.-L. Ingold, F. S. S. Rosa, N. B. Viana, H. M. Nussenzveig, and P. A. Maia Neto, “Probing the Casimir force with optical tweezers,” Europhys. Lett. 112, 44001 (2015).
[Crossref]

Popescu, M. N.

M. Elzbieciak-Wodka, M. N. Popescu, F. J. M. Ruiz-Cabello, G. Trefalt, P. Maroni, and M. Borkovec, “Measurements of dispersion forces between colloidal latex particles with the atomic force microscope and comparison with Lifshitz theory,” J. Chem. Phys. 140, 104906 (2014).
[Crossref]

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K. A. Milton, R. Guérout, G.-L. Ingold, A. Lambrecht, and S. Reynaud, “Negative Casimir entropies in nanoparticle interactions,” J. Phys. Condens. Matter 27, 214003 (2015).
[Crossref]

G.-L. Ingold, S. Umrath, M. Hartmann, R. Guérout, A. Lambrecht, S. Reynaud, and K. A. Milton, “Geometric origin of negative Casimir entropies: a scattering-channel analysis,” Phys. Rev. E 91, 033203 (2015).
[Crossref]

A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Thermal Casimir effect for Drude metals in the plane-sphere geometry,” Phys. Rev. A 82, 012511 (2010).
[Crossref]

A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Thermal Casimir effect in the plane-sphere geometry,” Phys. Rev. Lett. 104, 040403 (2010).
[Crossref]

A. Canaguier-Durand, P. A. Maia Neto, I. Cavero-Pelaez, A. Lambrecht, and S. Reynaud, “Casimir interaction between plane and spherical metallic surfaces,” Phys. Rev. Lett. 102, 230404 (2009).
[Crossref]

P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Casimir energy between a plane and a sphere in electromagnetic vacuum,” Phys. Rev. A 78, 012115 (2008).
[Crossref]

A. Lambrecht, P. A. Maia Neto, and S. Reynaud, “The Casimir effect within scattering theory,” New J. Phys. 8, 243 (2006).
[Crossref]

Rosa, F. S. S.

D. S. Ether, L. B. Pires, S. Umrath, D. Martinez, Y. Ayala, B. Pontes, G. R. de S. Araújo, S. Frases, G.-L. Ingold, F. S. S. Rosa, N. B. Viana, H. M. Nussenzveig, and P. A. Maia Neto, “Probing the Casimir force with optical tweezers,” Europhys. Lett. 112, 44001 (2015).
[Crossref]

Ruiz-Cabello, F. J. M.

M. Elzbieciak-Wodka, M. N. Popescu, F. J. M. Ruiz-Cabello, G. Trefalt, P. Maroni, and M. Borkovec, “Measurements of dispersion forces between colloidal latex particles with the atomic force microscope and comparison with Lifshitz theory,” J. Chem. Phys. 140, 104906 (2014).
[Crossref]

Scardicchio, A.

A. Scardicchio and R. L. Jaffe, “Casimir effects: an optical approach I. Foundations and examples,” Nucl. Phys. B 704, 552–582 (2005).
[Crossref]

R. L. Jaffe and A. Scardicchio, “Casimir effect and geometric optics,” Phys. Rev. Lett. 92, 070402 (2004).
[Crossref]

Schaden, M.

M. Schaden and L. Spruch, “Infinity-free semiclassical evaluation of Casimir effects,” Phys. Rev. A 58, 935–953 (1998).
[Crossref]

Sedighi, M.

M. Sedighi, V. B. Svetovoy, and G. Palasantzas, “Casimir force measurements from silicon carbide surfaces,” Phys. Rev. B 93, 085434 (2016).
[Crossref]

Somers, D. A. T.

J. L. Garrett, D. A. T. Somers, and J. N. Munday, “Measurement of the Casimir force between two spheres,” Phys. Rev. Lett. 120, 040401 (2018).
[Crossref]

Spreng, B.

B. Spreng, M. Hartmann, V. Henning, P. A. Maia Neto, and G.-L. Ingold, “Proximity force approximation and specular reflection: application of the WKB limit of Mie scattering to the Casimir effect,” Phys. Rev. A 97, 062504 (2018).
[Crossref]

Spruch, L.

M. Schaden and L. Spruch, “Infinity-free semiclassical evaluation of Casimir effects,” Phys. Rev. A 58, 935–953 (1998).
[Crossref]

Sushkov, A. O.

A. O. Sushkov, W. J. Kim, D. A. R. Dalvit, and S. K. Lamoreaux, “Observation of the thermal Casimir force,” Nat. Phys. 7, 230–233 (2011).
[Crossref]

Svetovoy, V. B.

M. Sedighi, V. B. Svetovoy, and G. Palasantzas, “Casimir force measurements from silicon carbide surfaces,” Phys. Rev. B 93, 085434 (2016).
[Crossref]

Tang, H. X.

D. Garcia-Sanchez, K. Y. Fong, H. Bhaskaran, S. Lamoreaux, and H. X. Tang, “Casimir force and in situ surface potential measurements on nanomembranes,” Phys. Rev. Lett. 109, 027202 (2012).
[Crossref]

Teo, L. P.

L. P. Teo, “Material dependence of Casimir interaction between a sphere and a plate: first analytic correction beyond proximity force approximation,” Phys. Rev. D 88, 045019 (2013).
[Crossref]

L. P. Teo, M. Bordag, and V. Nikolaev, “Corrections beyond the proximity force approximation,” Phys. Rev. D 84, 125037 (2011).
[Crossref]

Thornton, S.

G. Torricelli, I. Pirozhenko, S. Thornton, A. Lambrecht, and C. Binns, “Casimir force between a metal and a semimetal,” Europhys. Lett. 93, 51001 (2011).
[Crossref]

Torricelli, G.

G. Torricelli, I. Pirozhenko, S. Thornton, A. Lambrecht, and C. Binns, “Casimir force between a metal and a semimetal,” Europhys. Lett. 93, 51001 (2011).
[Crossref]

Trefalt, G.

M. Elzbieciak-Wodka, M. N. Popescu, F. J. M. Ruiz-Cabello, G. Trefalt, P. Maroni, and M. Borkovec, “Measurements of dispersion forces between colloidal latex particles with the atomic force microscope and comparison with Lifshitz theory,” J. Chem. Phys. 140, 104906 (2014).
[Crossref]

Umrath, S.

D. S. Ether, L. B. Pires, S. Umrath, D. Martinez, Y. Ayala, B. Pontes, G. R. de S. Araújo, S. Frases, G.-L. Ingold, F. S. S. Rosa, N. B. Viana, H. M. Nussenzveig, and P. A. Maia Neto, “Probing the Casimir force with optical tweezers,” Europhys. Lett. 112, 44001 (2015).
[Crossref]

S. Umrath, M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, “Disentangling geometric and dissipative origins of negative Casimir entropies,” Phys. Rev. E 92, 042125 (2015).
[Crossref]

G.-L. Ingold, S. Umrath, M. Hartmann, R. Guérout, A. Lambrecht, S. Reynaud, and K. A. Milton, “Geometric origin of negative Casimir entropies: a scattering-channel analysis,” Phys. Rev. E 91, 033203 (2015).
[Crossref]

Viana, N. B.

D. S. Ether, L. B. Pires, S. Umrath, D. Martinez, Y. Ayala, B. Pontes, G. R. de S. Araújo, S. Frases, G.-L. Ingold, F. S. S. Rosa, N. B. Viana, H. M. Nussenzveig, and P. A. Maia Neto, “Probing the Casimir force with optical tweezers,” Europhys. Lett. 112, 44001 (2015).
[Crossref]

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B. C. Berndt and B. P. Yeap, “Explicit evaluations and reciprocity theorems for finite trigonometric sums,” Adv. Appl. Math. 29, 358–385 (2002).
[Crossref]

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R. Zandi, T. Emig, and U. Mohideen, “Quantum and thermal Casimir interaction between a sphere and a plate: comparison of Drude and plasma models,” Phys. Rev. B 81, 195423 (2010).
[Crossref]

Adv. Appl. Math. (1)

B. C. Berndt and B. P. Yeap, “Explicit evaluations and reciprocity theorems for finite trigonometric sums,” Adv. Appl. Math. 29, 358–385 (2002).
[Crossref]

Appl. Phys. Lett. (1)

G. Bimonte, T. Emig, and M. Kardar, “Material dependence of Casimir forces: gradient expansion beyond proximity,” Appl. Phys. Lett. 100, 074110 (2012).
[Crossref]

Europhys. Lett. (3)

G. Torricelli, I. Pirozhenko, S. Thornton, A. Lambrecht, and C. Binns, “Casimir force between a metal and a semimetal,” Europhys. Lett. 93, 51001 (2011).
[Crossref]

D. S. Ether, L. B. Pires, S. Umrath, D. Martinez, Y. Ayala, B. Pontes, G. R. de S. Araújo, S. Frases, G.-L. Ingold, F. S. S. Rosa, N. B. Viana, H. M. Nussenzveig, and P. A. Maia Neto, “Probing the Casimir force with optical tweezers,” Europhys. Lett. 112, 44001 (2015).
[Crossref]

G. Bimonte, T. Emig, R. L. Jaffe, and M. Kardar, “Casimir forces beyond the proximity approximation,” Europhys. Lett. 97, 50001(2012).
[Crossref]

J. Chem. Phys. (1)

M. Elzbieciak-Wodka, M. N. Popescu, F. J. M. Ruiz-Cabello, G. Trefalt, P. Maroni, and M. Borkovec, “Measurements of dispersion forces between colloidal latex particles with the atomic force microscope and comparison with Lifshitz theory,” J. Chem. Phys. 140, 104906 (2014).
[Crossref]

J. Math. Phys. (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[Crossref]

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M. Bordag and V. Nikolaev, “Casimir force for a sphere in front of a plane beyond proximity force approximation,” J. Phys. A 41, 164002 (2008).
[Crossref]

J. Phys. Condens. Matter (1)

K. A. Milton, R. Guérout, G.-L. Ingold, A. Lambrecht, and S. Reynaud, “Negative Casimir entropies in nanoparticle interactions,” J. Phys. Condens. Matter 27, 214003 (2015).
[Crossref]

J. Stat. Mech. (1)

T. Emig, “Fluctuation-induced quantum interactions between compact objects and a plane mirror,” J. Stat. Mech. 2008P04007 (2008).
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Lect. Notes Phys. (2)

R. Decca, V. Aksyuk, and D. López, “Casimir force in micro and nano electro mechanical systems,” Lect. Notes Phys. 834, 287–309 (2011).
[Crossref]

S. K. Lamoreaux, “Progress in experimental measurements of the surface-surface Casimir force: electrostatic calibrations and limitations to accuracy,” Lect. Notes Phys. 834, 219–248 (2011).
[Crossref]

Nat. Phys. (1)

A. O. Sushkov, W. J. Kim, D. A. R. Dalvit, and S. K. Lamoreaux, “Observation of the thermal Casimir force,” Nat. Phys. 7, 230–233 (2011).
[Crossref]

New J. Phys. (1)

A. Lambrecht, P. A. Maia Neto, and S. Reynaud, “The Casimir effect within scattering theory,” New J. Phys. 8, 243 (2006).
[Crossref]

Nucl. Phys. B (1)

A. Scardicchio and R. L. Jaffe, “Casimir effects: an optical approach I. Foundations and examples,” Nucl. Phys. B 704, 552–582 (2005).
[Crossref]

Phys. Rev. A (5)

M. Schaden and L. Spruch, “Infinity-free semiclassical evaluation of Casimir effects,” Phys. Rev. A 58, 935–953 (1998).
[Crossref]

C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Derivative-expansion approach to the interaction between close surfaces,” Phys. Rev. A 89, 062120 (2014).
[Crossref]

A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Thermal Casimir effect for Drude metals in the plane-sphere geometry,” Phys. Rev. A 82, 012511 (2010).
[Crossref]

B. Spreng, M. Hartmann, V. Henning, P. A. Maia Neto, and G.-L. Ingold, “Proximity force approximation and specular reflection: application of the WKB limit of Mie scattering to the Casimir effect,” Phys. Rev. A 97, 062504 (2018).
[Crossref]

P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Casimir energy between a plane and a sphere in electromagnetic vacuum,” Phys. Rev. A 78, 012115 (2008).
[Crossref]

Phys. Rev. B (4)

R. Zandi, T. Emig, and U. Mohideen, “Quantum and thermal Casimir interaction between a sphere and a plate: comparison of Drude and plasma models,” Phys. Rev. B 81, 195423 (2010).
[Crossref]

C.-C. Chang, A. A. Banishev, R. Castillo-Garza, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Gradient of the Casimir force between Au surfaces of a sphere and a plate measured using an atomic force microscope in a frequency-shift technique,” Phys. Rev. B 85, 165443 (2012).
[Crossref]

M. Sedighi, V. B. Svetovoy, and G. Palasantzas, “Casimir force measurements from silicon carbide surfaces,” Phys. Rev. B 93, 085434 (2016).
[Crossref]

G. Bimonte, D. López, and R. S. Decca, “Isoelectronic determination of the thermal Casimir force,” Phys. Rev. B 93, 184434 (2016).
[Crossref]

Phys. Rev. D (6)

C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Proximity force approximation for the Casimir energy as a derivative expansion,” Phys. Rev. D 84, 105031 (2011).
[Crossref]

L. P. Teo, M. Bordag, and V. Nikolaev, “Corrections beyond the proximity force approximation,” Phys. Rev. D 84, 125037 (2011).
[Crossref]

G. Bimonte, “Classical Casimir interaction of perfectly conducting sphere and plate,” Phys. Rev. D 95, 065004 (2017).
[Crossref]

L. P. Teo, “Material dependence of Casimir interaction between a sphere and a plate: first analytic correction beyond proximity force approximation,” Phys. Rev. D 88, 045019 (2013).
[Crossref]

C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, “Derivative expansion for the electromagnetic Casimir free energy at high temperatures,” Phys. Rev. D 92, 125007 (2015).
[Crossref]

M. Bordag and I. Pirozhenko, “Vacuum energy between a sphere and a plane at finite temperature,” Phys. Rev. D 81, 085023 (2010).
[Crossref]

Phys. Rev. E (2)

G.-L. Ingold, S. Umrath, M. Hartmann, R. Guérout, A. Lambrecht, S. Reynaud, and K. A. Milton, “Geometric origin of negative Casimir entropies: a scattering-channel analysis,” Phys. Rev. E 91, 033203 (2015).
[Crossref]

S. Umrath, M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, “Disentangling geometric and dissipative origins of negative Casimir entropies,” Phys. Rev. E 92, 042125 (2015).
[Crossref]

Phys. Rev. Lett. (9)

G. Bimonte and T. Emig, “Exact results for classical Casimir interactions: Dirichlet and Drude model in the sphere-sphere and sphere-plane geometry,” Phys. Rev. Lett. 109, 160403 (2012).
[Crossref]

R. L. Jaffe and A. Scardicchio, “Casimir effect and geometric optics,” Phys. Rev. Lett. 92, 070402 (2004).
[Crossref]

T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, “Casimir forces between arbitrary compact objects,” Phys. Rev. Lett. 99, 170403 (2007).
[Crossref]

M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, “Plasma versus Drude modeling of the Casimir force: beyond the proximity force approximation,” Phys. Rev. Lett. 119, 043901 (2017).
[Crossref]

A. Canaguier-Durand, P. A. Maia Neto, I. Cavero-Pelaez, A. Lambrecht, and S. Reynaud, “Casimir interaction between plane and spherical metallic surfaces,” Phys. Rev. Lett. 102, 230404 (2009).
[Crossref]

A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, “Thermal Casimir effect in the plane-sphere geometry,” Phys. Rev. Lett. 104, 040403 (2010).
[Crossref]

J. L. Garrett, D. A. T. Somers, and J. N. Munday, “Measurement of the Casimir force between two spheres,” Phys. Rev. Lett. 120, 040401 (2018).
[Crossref]

D. Garcia-Sanchez, K. Y. Fong, H. Bhaskaran, S. Lamoreaux, and H. X. Tang, “Casimir force and in situ surface potential measurements on nanomembranes,” Phys. Rev. Lett. 109, 027202 (2012).
[Crossref]

A. A. Banishev, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, “Demonstration of the Casimir force between ferromagnetic surfaces of a Ni-coated sphere and a Ni-coated plate,” Phys. Rev. Lett. 110, 137401 (2013).
[Crossref]

Phys. Scr. (1)

M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, “Advancing numerics for the Casimir effect to experimentally relevant aspect ratios,” Phys. Scr. 93, 114003 (2018).
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[Crossref]

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V. A. Parsegian, Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists (Cambridge University, 2006).

M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir Effect (Oxford University, 2009).

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992).

W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge University, 2005).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (World Scientific, 2006).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds., NIST Digital Library of Mathematical Functions, Release 1.0.20, http://dlmf.nist.gov/6.2 .

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Figures (3)

Fig. 1.
Fig. 1. Sphere of radius R and plate separated by a distance L.
Fig. 2.
Fig. 2. The Fresnel plane for the incoming wave vector K(in) in general does not coincide with the scattering plane. The two planes are at an angle χ(in). The corresponding Fresnel plane for the outgoing wave vector K(out) is not shown.
Fig. 3.
Fig. 3. Specular reflection at (i) a tangent plane at the bottom of the sphere and (ii) at a slightly tilted tangent plane.

Tables (1)

Tables Icon

Table 1. Relative Contribution of the Terms Arising from Diffraction and Geometrical Optics for Transverse Electric (TE) and Transverse Magnetic (TM) Polarizations to the Total Correction of Order 1/R

Equations (83)

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

Kz=ϕkz
kz=(ω2c2k2)1/2.
ϵ^TE=z^×K^|z^×K^|,ϵ^TM=ϵ^TE×K^,
x,y,z|ω,k,p,ϕ=ϵ^p(12π|ωckz|)1/2exp[i(k·r+ϕkzz)],
ξ2=c2(κ2k2).
E=0dξ2πtrlog(1M(ξ)).
M=TPSRSTSPRP.
E=r=11r0dξ2πtrMr,
trMr=p0,,pr1dk0dkr1(2π)2rj=0r1e2κj(L+R)rpj×kj+1,pj+1,|RS|kj,pj,+,
ϵ^=K^(out)×K^(in)|K^(out)×K^(in)|,ϵ^(in)=ϵ^×K^(in),ϵ^(out)=ϵ^×K^(out).
K(out),|RS|K(in),=2πcξκ(out)S,K(out),|RS|K(in),=2πcξκ(out)S,
S==12+1(+1)[aπ(cosΘ)+bτ(cosΘ)],
S==12+1(+1)[aτ(cosΘ)+bπ(cosΘ)].
cos(Θ)=c2ξ2(κ(in)κ(out)+k(in)·k(out)).
k(out),TM,|RS|k(in),TM,+=2πcξκ(out)(AS+BS),
k(out),TE,|RS|k(in),TE,+=2πcξκ(out)(AS+BS),
k(out),TM,|RS|k(in),TE,+=2πcξκ(out)(CS+DS),
k(out),TE,|RS|k(in),TM,+=2πcξκ(out)(CS+DS).
A=cos(χ(out))cos(χ(in)),B=sin(χ(out))sin(χ(in)),C=sin(χ(out))cos(χ(in)),D=cos(χ(out))sin(χ(in)),
cos(χ(in))=ϵ^TE(K(in))·ϵ^,cos(χ(out))=ϵ^TE(K(out))·ϵ^.
A=1,B=C=D=0.
Sp=SpWKB(1+1Rsp+O(R2)).
SpWKB=(1)pξR2cexp[2ξRcsin(Θ2)],
s=c2ξcos(Θ)sin3(Θ/2),s=c2ξ1sin3(Θ/2).
k(out),p(out),|RS|k(in),p(in),+πRκ(out)exp[2ξRcsin(Θ2)]ρp(out),p(in)
ρTM,TM=(AB)+1R(AsBs),ρTE,TE=(AB)1R(AsBs),ρTE,TM=(CD)+1R(CsDs),ρTM,TE=(CD)+1R(CsDs).
E=EPFA(1+β1LR+o(R1)).
EPFA=cπ3R720L2
β1=βd+βgo,
trMr(R4π)rdk0dkr1g(k0,,kr1)eRf(k0,,kr1),
g(k0,,kr1)=p0,,pr1j=0r1(1)pje2κjLκjρpj+1,pj.
f(k0,,kr1)=j=0r1ηj,j+1,
ηj,j+1=κj+κj+1[2(ξ2c2+κjκj+1+kj·kj+1)]1/2.
k0==kr1ksp,
H=(Hxx00Hyy)
(Hxx)ij=2fki,xkj,x|sp
Γr=(21112111112),
Γ2=(1111),
kj,x=l=0r1Wjlvl,x
Wjl=1rexp(2πirjl).
(WTHxxW)jl=λjδj,rl
λj=2κspsin2(πjr)
trMr=R2rξ/cdκspκspr[F0+1RF1+o(R1)].
j=1r11λj=(2κsp)r1r2.
F0=g|sp
F1=g|sp(ijk2fijkfi¯j¯k¯+3fijj¯fi¯kk¯24λiλjλkijfii¯jj¯8λiλj)+ijgifi¯jj¯2λiλj+igii¯2λi.
gvi,x|sp=l=0r1Wilgkl,x|sp,
F1=g|sp(ijkfijkfi¯j¯k¯12λiλjλkijfii¯jj¯8λiλj)+igii¯2λi.
g|sp=gTE+gTM,
gp=exp(2κspLr)κspr(1+rRsp|sp).
sTE|sp=1κsp3(ξ22c2κsp2),sTM|sp=1κsp3ξ22c2.
(trMTEr)0=RLeu4r2+18[(u24)E1(u)(u1)eu]
(trMTMr)0=RLeu4r218[u2E1(u)(u1)eu].
Ep,0=EPFA(12+βd,pLR)
βd,TE=252π2,
βd,TM=52π2.
βd=15π2.
ρTM,TM=ρTE,TE=cos(χ),ρTE,TM=ρTM,TE=sin(χ).
p0,,pr1j=0r1(1)pjρpj1,pj=i=0r1cos(χi+1,i)(1j>l=0r1tan(χj+1,j)tan(χl+1,l)),
(2vi,xvri,xp0,,pr1j=0r1(1)pjρpj1,pj)|sp=j(χj+1,jvi,xχj+1,jvri,x)|spj>l(χj+1,jvi,xχl+1,lvri,x+χj+1,jvri,xχl+1,lvi,x)|sp,
(trMpr)1=(r21)exp(2Lrξ/c)12r2,
βgo=135π2,
β1=1320π21.693.
βTE=1615π21.353,
βTM=165π20.339.
βDD=16
βNN=1620π2.
F1=g|sp(D112D28)+D32
D1=α,β,γ{x,y}i,j,l=1r11λiλjλl3fvi,αvj,βvl,γ3fvi¯,αvj¯,βvl¯,γ,D2=α,β{x,y}i,j=1r11λiλj4fvi,αvi¯,αvj,βvj¯,β,D3=α{x,y}i=1r11λi2gvi,αvi¯,α,
D1=p,q=0r1m,n,s=pp+1t,u,w=qq+1a(mt)a(nu)a(sw)×dpq(m,n,s;t,u,w),
a(s)=1rj=1r1e2πijs/rλj
dpq(m,n,s;t,u,w)=α,β,γ{x,y}3ηp,p+1km,αkn,βks,γ3ηq,q+1kt,αku,βkw,γ.
j=1r1e2πijs/rsin2(πj/r)=13(r26|s|r+6s21),
a(s)=κsp6r(r26sr+6s21).
dpq(p,p,p;q,q,q)=d,dpq(p+1,p,p;q,q,q)=dpq(p,p,p;q+1,q,q)=d3,dpq(p+1,p,p;q+1,q,q)=d3
d=34ksp2κsp6.
dpq(p+1,p,p;q,q+1,q)=dpq(p+1,p,p;q,q,q+1)=0.
D1=p,q=0r1A(pq)
A(s)=d(6a3(s)+[a(s1)+a(s+1)]×[a(s1)a(s+1)4a2(s)]).
D1=(r2)(r1)2(c2κsp2ξ2)rc2κsp3.
D2=2(r1)2((r2)c2κsp23rξ2)3rc2κsp3
D3=(r21)(ξ2+Lκsp(c2κsp2+ξ2))3c2κsp3g|sp.
F1=(r21)(rLκsp(c2κsp2+ξ2)+ξ2)6rc2κsp3g|sp.