What are Nonlinear Optics - 2nd Order Processes

[Greg Bernhardt]

Definition/Summary

Nonlinear optical processes that occur due to the presence of a second-order nonlinear susceptibility are termed 2nd order processes, or three-wave mixing processes. There are four second order processes, second harmonic generation, sum and difference frequency generation and optical rectification.

Crystals commonly used for inducing second order nonlinear effects are Lithium Niobate, Beta-Barium Borate (BBO) and Mono-Potassium Phosphate (KDP).

Equations

Optical Rectification.

$$E_1^2+E_2^2$$

Second Harmonic Generation.

$$E_{1}^2e^{(i(2\omega_1 t-2k_{1}r))}+E_{2}^2e^{(i(2\omega_2 t-2k_{2}r))}+c.c.$$

Sum Frequency Generation.

$$E_{1}E_{2}e^{(i((\omega_1+\omega_2)t-(k_{1}+k_{2})r)}+c.c.$$

Difference Frequency Generation.

$$E_{1}E_{2}^*e^{(i((\omega_1-\omega_2)t-(k_{1}-k_{2})r)}+c.c.$$

Extended explanation

The 2nd order nonlinear polarisation is given by;

$$P_{NL} = \epsilon_0\chi^{(2)}\mathbf{E.E}$$

Note that there are three waves present, two "pump" waves and a third "response" wave, $P_{NL}$. For this reason, second order nonlinear optical processes are sometimes referred to as "3-wave mixing processes". Assume that the pump fields are of the form $\frac{1}{2}\mathbf{E_1}e^{(i\omega_1 t-\mathbf{k_1.r})}+ c.c.$ and $\frac{1}{2}\mathbf{E_2}e^{(i\omega_2 t-\mathbf{k_2.r})} +c.c.$, then;

$$P_{NL} = \epsilon_0\chi^{(2)}(\mathbf{E_1}exp(i(\omega_1 t-\mathbf{k_1.r}))+\mathbf{E_2}e(i(\omega_2 t-\mathbf{k_2.r}))+ c.c.)^2$$

For clarity, let us look at one component of $P_{NL}$.

$$P_{NLi} \alpha (E_{1}^2+E_{2}^2+E_{1}^2e^{(i(2\omega_1 t-2k_{1}r))}+E_{2}^2e^{(i(2\omega_2 t-2k_{2}r))}+E_{1}E_{2}e^{(i((\omega_1+\omega_2)t-(k_{1}+k_{2})r)}+E_{1}E_{2}^*e^{(i((\omega_1-\omega_2)t-(k_{1}-k_{2})r)}+c.c.)$$

the overall value of $P_{NLi}$ will depend on the components of the nonlinear susceptibility $\chi^{(2)}$. The different terms represent different second order nonlinear processes;

Optical Rectification.

$$E_1^2+E_2^2$$

Optical rectification is the appearance of a DC potential difference across a 2nd order nonlinear medium in the presence of an optical field.

Second Harmonic Generation.

$$E_{1}^2e^{(i(2\omega_1 t-2k_{1}r))}+E_{2}^2e^{(i(2\omega_2 t-2k_{2}r))}+c.c.$$

Each pump field will generate a second harmonic that is twice the frequency (or equivalently, half the wavelength) of the original field. The evolution of the second harmonic depends on the phase matching - that is whether the second harmonic is generated in phase with the pump field.

Sum Frequency Generation.

$$E_{1}E_{2}e^{(i((\omega_1+\omega_2)t-(k_{1}+k_{2})r)}+c.c.$$

Two pump fields will interact in a second order nonlinear medium, producing a field whose frequency is the sum of the original pump fields. The reverse of this process, parametric down-conversion, is widely used in tunable laser sources.

Difference Frequency Generation.

$$E_{1}E_{2}^*e^{(i((\omega_1-\omega_2)t-(k_{1}-k_{2})r)}+c.c.$$

Two pump fields will also produce a field whose frequency is the difference of the original pump fields.

Note too, that these processes can be cascaded, that is, a second harmonic can generate its own second harmonic (i.e. the fourth harmonic of the original beam) and so on. The waves that emerge from a second order medium will be determined by the phase matching conditions - those processes that are not properly phase matched will be suppressed.

Phase Matching

To achieve efficient conversion of energy from the pump fields to the fields produced via second-order interactions, phase-matching conditions must be met. This condition arises because the second harmonic (for example) will not be in phase with the pump field due to material dispersion (the medium possessing a different refractive index for different wavelengths).

To achieve phase-matching, a birefringent medium is often used. Uniaxial birefringent media have two refractive indices, essentially one for each orthogonal polarisation. In such a medium, the pump refractive index can be matched to the refractive index of the second harmonic (whose polarisation is orthogonal to the pump). An alternative method of phase matching (quasi phase-matching) is to use crystals whose second order coefficient reverses sign every half wavelength. Such crystals are termed periodically poled crystals.

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[APhysicist]

Thanks for the detailed explanation of 2nd order processes and phase matching! I appreciate the equations and extended explanation which make it easier to understand this concept.