Parametric Down-Conversion of Photons

 

Nonlinear Optics

One incoming photon is converted into two lower-energy photons, typically in a nonlinear optical medium. Energy is conserved: $E_{\text{in}}=E_1+E_2$.

Parametric Down-Conversion

A nonlinear optical process in which one pump photon is converted into two lower-energy photons, known as the signal and idler photons.

1

Energy Conservation

The energy of the pump photon equals the sum of the energies of the signal and idler photons.

\[ \boxed{\hbar\omega_p=\hbar\omega_s+\hbar\omega_i} \] \[ \boxed{\omega_p=\omega_s+\omega_i} \]

ωₚ

Pump angular frequency

ωₛ

Signal angular frequency

ωᵢ

Idler angular frequency

ℏ

Reduced Planck constant

Wavelength Relationship

Using \(E=hc/\lambda\), energy conservation can also be expressed in terms of vacuum wavelengths.

\[ \boxed{ \frac{1}{\lambda_p} = \frac{1}{\lambda_s} + \frac{1}{\lambda_i} } \]

2

Momentum Conservation

Efficient down-conversion requires phase matching. In a common wave-vector description, the pump wave vector equals the vector sum of the signal and idler wave vectors.

\[ \boxed{\mathbf{k}_p=\mathbf{k}_s+\mathbf{k}_i} \]

Here, kₚ, kₛ and kᵢ are the wave vectors of the pump, signal and idler fields, respectively. The phase-matching condition depends on the optical properties of the nonlinear medium.

3

Degenerate Down-Conversion

In degenerate parametric down-conversion, both emitted photons have the same frequency and, consequently, the same vacuum wavelength.

\[ \omega_s=\omega_i=\frac{\omega_p}{2} \] \[ \boxed{\lambda_s=\lambda_i=2\lambda_p} \]

Example

Suppose the pump photon has a vacuum wavelength of 400 nm. In the degenerate case, each emitted photon has a wavelength of 800 nm.

Pump photon

400 nm

→

Signal

800 nm

Idler

800 nm

Summary

In parametric down-conversion, the pump photon is converted into two photons through a nonlinear optical interaction. It is not ordinary atomic absorption followed by emission. Energy is conserved, and efficient conversion requires appropriate phase-matching conditions.