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Table of contents
Introduction
I. Reminder on the use of the scattering formalism for Casimir computations, and example of the plane-plane geometry
1. The Scattering formula
1.1. The planar electromagnetic modes
1.2. The cavity operator
1.3. The Casimir energy from the radiation pressure
1.4. The example of the plane-plane geometry
2. Optical properties of materials
2.1. The dielectric function
2.2. The plasma model
2.3. The Drude model
2.4. Dielectrics
2.5. The example of the plane-plane geometry
3. The scattering formula at non-zero temperature
3.1. The Matsubara sum
3.2. The example of the plane-plane geometry
4. The proximity force approximation (PFA) 55
4.1. The PFA formula in the sphere-plane geometry
4.2. The primo-potential D
4.3. Beyond the PFA method
II. Method for the Casimir effect in the sphere-plane geometry
5. The scattering formula in the sphere-plane geometry
5.1. Derivation of the scattering operator D({)
5.2. Explicit form of the various involved quantities
5.3. Transformation to real and simpler quantities
5.4. Conclusion
6. Numerical issues
6.1. From operators to matrices: the truncation to `max
6.2. Differentiations with respect to L and T
6.3. Integration/sum over ~
6.4. Integration over cos
6.5. Modified Bessel functions
6.6. Numerical stability
6.7. Computation of the zero-frequency term
III. Analytical limits
7. Low-frequency limit
7.1. Mie coefficients
7.2. Fresnel coefficients
7.3. Spherical harmonics and finite rotations
7.4. Integration over cos
7.5. Determinant of the scattering matrix
7.6. Conclusion: the first Matsubara term and the high-temperature limit
8. Long-distance limit
8.1. The dipolar-simple scattering approximation
8.2. Perfect mirrors
8.3. Metallic scatterers modelled with the plasma model
8.4. Metallic scatterers modelled with the Drude model
8.5. Dielectric scatterers
IV. Results at zero temperature
9. Beyond-PFA computations in the literature
9.1. Ways to measure the accuracy of PFA
9.2. Scalar results for perfect mirrors
9.3. Electromagnetic results for perfect mirrors
9.4. Experimental prescription on and conclusion
10.Results for perfect mirrors
10.1. Behaviour of numerical results at short and large separation
10.2. Power laws
10.3. Beyond-PFA corrections
11.Results for metallic mirrors
11.1. Observation of the effect of imperfect reflection
11.2. Power laws
11.3. Correlations between the effects of finite conductivity and geometry
11.4. Influence of conductivity on the beyond-PFA corrections
12.Results for dielectric nanospheres
12.1. The Casimir-Polder formula for a dielectric nanosphere
12.2. The complete multipolar expression E
12.3. Averaging Casimir-Polder over the sphere’s volume
V. Results at non-zero temperature
13.Perfect mirrors at ambient temperature
13.1. Observation of the thermal effects
13.2. Power laws
13.3. Correlations between curvature and thermal effects
13.4. Casimir entropy
13.5. Comparison with PFA at short distance
14.Metallic mirrors at ambient temperature
14.1. Influence of temperature for metallic materials
14.2. Influence of imperfect reflection at ambient temperature
14.3. Study of various interplays
14.4. Beyond-PFA corrections
14.5. Power laws
15.High-temperature regime
15.1. Perfect mirrors
15.2. Drude model for metallic mirrors
15.3. Ratio of perfect mirrors result over Drude metals result
Conclusion and outlook
Appendix
A. Proofs of lemmas, properties and theorems
A.1. Lemma 1 (p.48)
A.2. Lemma 2 (p.48)
A.3. Property 4 (p.48)
A.4. Property 5 (p.75)
A.5. Property 6 (p.86)
A.6. Property 12 (p.101)
A.7. Property 13 (p.114)
A.8. Property 14 (p.128)
B. Approximations methods for the sphere-plane geometry
B.1. PFA methods
B.2. PWS methods
Published articles
Casimir Interaction between Plane and Spherical Metallic Surfaces (PRL, 2009)
Thermal Casimir Effect in the Plane-Sphere Geometry (PRL, 2010)
Thermal Casimir effect for Drude metals in the plane-sphere geometry (PRA, 2010)
Casimir interaction between a dielectric nanosphere and a metallic plane (PRA, 2011)
References
