Second Harmonic Generation in the Frequency Domain

The emission spectra from different types of laser system cover a large part of the visible and near visible part of the electromagnetic spectrum. However, it is still more difficult to generate laser emission in the short-wavelength part of the spectrum than in the long-wavelength part. To circumvent this dilemma, it is common to use nonlinear frequency mixing to generate new wavelengths from the existing laser wavelengths. A common approach is to start with a Nd:YAG laser that emits at 1 064 nm wavelength and then frequency-double that wavelength to green at 532 nm. Given those two wavelengths, it is also possible to mix them, which results in the generation of light in ultraviolet (UV) at 355 nm — an effective frequency tripling of the original wavelength at 1 064 nm.

This model demonstrates how two frequency-domain interfaces can be coupled together to simulate the second harmonic generation process, where light from the fundamental wavelength (frequency) is injected in a nonlinear crystal that generates the second harmonic frequency, which is twice the fundamental frequency. The results are compared with analytical results obtained within the Slowly Varying Envelope Approximation (SVEA).

Model Definition

The geometry for the model is very simple, consisting only of a slender two-dimensional rectangle. The rectangle is many wavelengths long in the propagation direction, but consists of only one mesh element in the direction orthogonal to the propagation direction.

The first Electromagnetic Waves, Frequency Domain interface is defined for the fundamental frequency f1 and the second Electromagnetic Waves, Frequency Domain interface is defined for the second harmonic frequency 2f1.

The only incident wave is polarized in the y-direction and launched at the fundamental frequency using a Scattering Boundary Condition feature.

The two interfaces are coupled using a Polarization feature added to each of the interfaces. For the fundamental interface, the polarization is given by

(1)

and for the second harmonic interface the polarization is given by

(2)

where d is a nonlinear coefficient for the process, E1y is the y-component of the electric field at the fundamental frequency, and E2y is the y-component of the electric field at the second harmonic frequency. For more details regarding second harmonic generation, see for example Ref. 1.

The results from the simulation are compared with the analytical results obtained within the Slowly Varying Envelope Approximation (SVEA) (see Ref. 1). The analytical results for the photon flux density, assuming perfect phase matching, are for the fundamental wave

(3)

and for the second harmonic wave

(4)

where

is the incident photon flux density for the fundamental wave, the constant ϒ is defined by

(5)

Z0 is the characteristic impedance of the medium, ω is the angular frequency, and

is the incident intensity for the fundamental wave. As seen from Equation 3 and Equation 4, when x goes to infinity the photon flux density for the fundamental goes to zero, whereas the photon flux density for the second harmonic approaches half of the initial fundamental photon flux density. Since the photon energy for the second harmonic is twice that of the fundamental, the energy is conserved in the process.

Results and Discussion

Figure 1 shows the y-component of the electric field of the fundamental wave. As shown, the amplitude decreases when the wave propagates through the medium and energy is transferred to the second harmonic wave. Figure 2 shows the y-component of the electric field for the second harmonic wave. For this wave, the initial amplitude is zero. Upon propagation, the amplitude increases, as energy is transferred from the fundamental wave. Notice also that the wavelength for the second harmonic field is half that of the fundamental wave.

Figure 1: The electric field distribution of the fundamental wave (y-component).

Figure 2: The electric field distribution of the second harmonic wave (y-component).

Figure 3 shows a comparison of the fundamental and the second harmonic, confirming the conclusions drawn from the comparison of the two previous figures.

Figure 3: A comparison between the y-components of the electric fields for the fundamental and the second harmonic wave.

Finally, Figure 4 compares the results from the simulation with analytical results obtain by applying the Slowly Varying Envelope Approximation (SVEA) (see Equation 3 and Equation 4 in the Model Definition section above). As the energy for each photon in the second harmonic wave is twice that of the energy of the photons in the fundamental wave, the curves indicate that the energy is conserved in the second harmonic generation process.

Figure 4: The photon flux density (in units of photons per m2 and s) for the fundamental and the second harmonic wave. The diamonds represent the simulated results (blue diamonds representing the fundamental wave and green diamonds representing the second harmonic wave), whereas the red line represents the analytical result in Equation 3 and the cyan line represents the analytical results in Equation 4.

Notes About the COMSOL Implementation

The value for the nonlinear coefficient (d = 1×10-18 C/V2) in this proof of concept model is intentionally chosen to be unphysically large, to allow for a small simulation domain. More typical values for realistic nonlinear materials are of the order of d = 1×10-24–1×10‑21 C/V2.

As mentioned in Ref. 1, there exists different conventions for expressing the nonlinear coefficients in the series expansion of the polarization in terms of the electric field. In this model, the second order nonlinear coefficient follows the convention in Ref. 1.

Reference

1. B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics, John Wiley & Sons, chap. 19, 1991.

Wave_Optics_Module/Verification_Examples/second_harmonic_generation_frequency_domain

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Parameters 1

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Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	1	1	Refractive index
Refractive index, imaginary part	ki_iso ; kiii = ki_iso, kiij = 0	0	1	Refractive index