CritPt / Challenge_18_main / Two dielectric nanoparticles are deeply trapped in two Gaussian optical traps…

# Problem Setup

Two dielectric nanoparticles are deeply trapped in two Gaussian optical traps that propagate along the z-axis, both characterized by the wave vector k and the Rayleigh range z_(R). Suppose the focal planes of these traps are located at z=0, and the nanoparticles are located at z=z₁ and z=z₂ respectively, where z₁,z₂≪z_(R). Let the distance between the two nanoparticles be d, satisfying the far-field condition kd≫1. The polarizabilities of the two nanoparticles are α₁ and α₂, respectively. Both the tweezers have identical polarization, the electric field amplitudes are E₁ and E₂, and the phases at the focal planes are ϕ₁ and ϕ₂, respectively.

# Main problem:
Assume that, at equilibrium, the distance vector between the two spheres is d₀=(d₀,0,0). The angle between the laser polarization and the particle-connecting axis is π/2. Derive k₁ and k₂ in the following equations of motion along the z-direction for the two nanospheres:

$$\begin{aligned}
m{{\ddot z}_1} =  - m\Omega _1^2{z_1} - ({k_1} + {k_2}){z_1} + ({k_1} + {k_2}){z_2},\\
m{{\ddot z}_2} =  - m\Omega _2^2{z_2} - ({k_1} - {k_2}){z_2} + ({k_1} - {k_2}){z_1}.
\end{aligned}$$

# Problem Setup

Two dielectric nanoparticles are deeply trapped in two Gaussian optical traps that propagate along the $z$-axis, both characterized by the wave vector $k$ and the Rayleigh range $z_R$. Suppose the focal planes of these traps are located at $z=0$, and the nanoparticles are located at $z=z_1$ and $z=z_2$ respectively, where ${z_1},{z_2} \ll {z_R}$. Let the distance between the two nanoparticles be $d$, satisfying the far-field condition $kd \gg 1$. The polarizabilities of the two nanoparticles are $\alpha_1$ and $\alpha_2$, respectively. Both the tweezers have identical polarization, the electric field amplitudes are $E_1$ and $E_2$, and the phases at the focal planes are $\phi_1$ and $\phi_2$, respectively.

# Main problem:
Assume that, at equilibrium, the distance vector between the two spheres is $d_0 = (d_0,0,0)$. The angle between the laser polarization and the particle-connecting axis is $\pi/2$. Derive $k_1$ and $k_2$ in the following equations of motion along the $z$-direction for the two nanospheres:

$$\begin{aligned}
m{{\ddot z}_1} =  - m\Omega _1^2{z_1} - ({k_1} + {k_2}){z_1} + ({k_1} + {k_2}){z_2},\\
m{{\ddot z}_2} =  - m\Omega _2^2{z_2} - ({k_1} - {k_2}){z_2} + ({k_1} - {k_2}){z_1}.
\end{aligned}$$