When light travels through a material, it interacts with it on an atomic and molecular level. You can think of these atoms or molecules as arrays of dipoles. The electric field from the incident light drives these dipoles causing them to oscillate like springs as it travels through the material.
In most cases, the light will be unaffected and have exactly the same frequency when it leaves the medium. However, it is possible for the light to force these dipoles to the point that they oscillate with a nonlinear response such that the re-emitted light contains additional frequencies, like the harmonics on a spring. Some materials are more prone to exhibit second order nonlinear or χ (2) responses, others can be more susceptible to third-order or χ (3) responses. The type of nonlinear response depends wholly on the structure of the material.
Second order nonlinear frequency conversion
Second order nonlinear processes involve the mixing of three electromagnetic waves, where the magnitude of the nonlinear response of the crystal is characterized by the χ (2) coefficient. This can give rise to the following interactions:
- Difference frequency generation (DFG)
- Second harmonic generation (SHG)
- Sum frequency generation (SFG)
Second harmonic generation (SHG), or frequency doubling, is the most common application that utilizes the χ (2) properties of a nonlinear crystal. In SHG, two input pump photons with the same wavelength λ p are combined through a nonlinear process to generate a third photon at λSHG, where,
OR, in terms of frequency,
Similar to SHG, sum frequency generation (SFG) combines two input photons at λ p and λs to generate an output photon at λ SFG , where,
OR, in terms of frequency,
Alternatively, in difference frequency generation (DFG) when two input photons at λp and λs are incident on the crystal, the presence of the lower frequency signal photon, λs, stimulates the pump photon, λp, to emit a signal photon λ s and idler photon at λ i , where,
OR, in terms of frequency,
In this process, two signal photons and one idler photon exit the crystal resulting in an amplified signal field. This is known as optical parametric amplification. Furthermore, by placing the nonlinear crystal within an optical resonator, also known as an optical parametric oscillator (OPO), the efficiency can be significantly enhanced.
Phase Matching
In all of these processes, photon energy is conserved; however in order for any of these the second order nonlinear frequency conversion interactions to occur, momentum must also be conserved. This is otherwise known as phase matching.
Phase matching refers to fixing the relative phase between two or more frequencies of light as they propagate through the crystal, such as frequency doubling, sum and difference frequency generation . The refractive index is dependent on the frequency of light. Thus, the phase relation between two photons of different frequencies will vary as the photons propagate through the material, unless the crystal is phase matched for those frequencies. It is necessary for the phase relation between the input and generated photons to be maintained throughout the crystal for efficient nonlinear frequency conversion of input photons. If this is not the case, the generated photons will move in and out put phase with each other in a sinusoidal manner, limiting the number of generated photons that exit the crystal. This is shown in the figure below. Traditional phase matching requires that the light is propagated through the crystal in a direction where the natural birefringence of the crystal matches the refractive index of the generated light. Despite providing perfect phase matching, this technique is limited to a small range of wavelengths in those materials that can be phase matched.
PPLN is an engineered, quasi-phase-matched material. The term engineered refers to the fact that the orientation of the lithium niobate crystal is periodically inverted (poled). By inverting the crystal orientation at every peak of the sinusoidal generation, one can avoid the photons slipping out of phase with each other. As a result, the number of generated photons will grow as the light propagates through the PPLN, yielding a high conversion efficiency of input to generated photons (see above figure).
The period with which the crystal needs to be inverted (the poling period) depends on the interacting wavelengths and the temperature of the PPLN. For example, a PPLN crystal with a poling period of 6.6μm will efficiently generate frequency doubled photons from 1060nm photons when the crystal temperature is held at 100°C. By increasing the temperature of the crystal to 200°C the same PPLN crystal will efficiently generate frequency doubled photons from 1068.6nm wavelength photons. Thus, changing the temperature of the crystal therefore varies the phase matching conditions, allowing some tuning of the wavelength interaction.
