Atomic-Molecular-Optical Physics

Cauchy (1836) \[ n^2 = a + \frac{b}{\lambda^2} + \frac{d}{\lambda^4} + \frac{f}{\lambda^6} + \cdots. \]

Sellmeier (1872) Formulated after the discovery of anomalous dispersion by Christiansen and Kundt, to account for the anomalies. \[ n^2 - 1 = \sum_k \frac{a_k\lambda^2}{\lambda^2 - \lambda_k^2}. \]

Helmholtz (1875) Helmholtz hypothesized the damping of the ether. \[ n^2 - 1 = \sum_k \frac{a_k\lambda^2}{\lambda^2 - \lambda_k^2 + b_k\lambda^2}. \] Although the original motivation is now proven to be wrong, the term itself remains to be important.

Lorentz and Drude The molecules within matter is modelled as harmonic oscillators.

As a plane electromagnetic wave \( E_s(z,t) = E_0\exp(i\omega (t - z/c)) \) passees through a thin dispersive medium, the retarded wave

\begin{align*} E'(z,t) &= E_s(z,t - \Delta t) \\ &= \exp(-i\omega\Delta t) E_s(z,t) \\ &\approx (1 - i\omega \Delta t) E_s(z,t) \\ & = \left( 1 - \frac{i\omega(n-1) \Delta z}{c} \right)E_s(z,t) \\ \end{align*}

is due to the superposition with the wave \( E_q(z,t) \) generated by the oscillating molecules

\begin{align*} E'(z,t) &= E_s(z,t) + E_q(z,t) \\ &= E_s(z,t) - \frac{i\omega\eta q x_0}{2\varepsilon_0 c} \exp(i\omega(t - z/c)). \end{align*}

Thus, we find that the refractive index is \[ n - 1 = \frac{qx_0N}{2\varepsilon_0E_0}. \]

Let us model the molecules with a forced damped oscillator \[ m \ddot{x} = - \omega_0^2x - \gamma \dot{x} + qE_0e^{i\omega t} \] the solution would have the amplitude \[ x_0 = \frac{qE_0}{m}\frac{1}{\omega_0^2 - \omega^2 + i\gamma\omega}. \]

Plugging \( x_0 \) into the equality we drived earlier, the formula for the index of refraction is obtained: \[ n-1= \frac{e^2N}{2\varepsilon_0m}\frac{1}{\omega_0^2 - \omega^2 + i\gamma\omega}. \]

The imaginary part in this formula constitute the decay of the electromagnetic field.

Lorenz-Lorentz Equation \[ \frac{n^2 - 1}{n^2 + 2} = \frac{4 \pi k_e}{3}N\chi \] where \( \chi \) is the polarizability.

Ladenburg and Reiche (1923) \[ \mathbf{P}_k = \frac{c^3 \mathbf{E}}{32 \pi^4 k_e} \sum_{j> k} \frac{A_{jk}}{\nu_{jk}^2(\nu_{jk}^2 -\nu^2)} \]

Kramers (1924) \[ \mathbf{P}_k = \frac{c^3 \mathbf{E}}{32 \pi^4 k_e} \left[ \sum_{j> k} \frac{A_{jk}}{\nu_{jk}^2(\nu_{jk}^2 -\nu^2)} - \sum_{j < k} \frac{A_{kj}}{\nu_{kj}^2(\nu_{kj}^2 -\nu^2)} \right] \]

• 형광

The electron decays from an excited singlet state to singlet state.

• 야광

Delayed light emission after excitation

The electron is excited to a singlet state and moves to a triplet state via intersystem crossing. The decay to ground singlet state is forbidden and much slower. It was first observed in phosphorus, hence the name.

Radiation exposure cause photoluminescence