**PPLN Guide: Principle****Introduction of nonlinear wavelength conversion****Light-matter interaction**When light propagates in a dielectric medium, the electric field (

**E**) of light interacting with the dielectric medium will induce some amount of dielectric polarization (**P**) in the dielectric medium. Here, the dielectric medium with its atoms/molecules as dipole arrays acting like a spring will respond to electric field**(E)**and generate induced dielectric polarization**(P)**. The induced dielectric polarization**(P)**describes the overall response behavior of a dielectric medium to electric field**(E)**, such as reflection, refraction, dispersion, and nonlinear harmonic wave generation (as illustrated in figure 1). In linear dielectric medium, the response is linear (as shown in figure 2-a) and the frequency of light will not be altered; in nonlinear dielectric medium, the response is nonlinear (as shown in figure 2-b) and the light will force those dipoles to oscillate with a nonlinear response such that the re-emitted light contains additional frequencies, which become the sources of nonlinear harmonic wave generation.Figure 1: Dipole arrays of a dielectric medium act like a spring, which responds to electric field (

**E**) and generates induced dielectric polarization (**P**).Figure 2-a (left): Linear dielectric polarization. Figure 2-b (right): Nonlinear dielectric polarization.

In the above figure,

**X**^{(1)}is linear susceptibility, while**X**and^{(2) }**X**are the second-order and third-order nonlinear susceptibility. The linear term^{(3)}**X**determines the linear propagation of optical waves, such as refraction, reflection, diffraction, and dispersion; while the higher order terms^{(1)}**X**and^{(2)}**X**correspond to nonlinear effects for harmonics wave generation under strong electric field. The second order term vanishes in dielectric mediums with inversion symmetry structures but could be large in several nonlinear mediums with non-centrosymmetric crystal structures, such as Lithium Niobate (LiNbO^{(3)}_{3}).Additional Note for dielectric polarization (P) vs. electric field (E):

**Phase-matching**To enable efficient nonlinear wavelength conversion, both photon energy conservation and photon momentum conservation are to be satisfied simultaneously in the nonlinear optical process, as shown in figure 3. The requirement on the photon momentum conservation is also called “phase-matching” constraint. If such a phase matching condition is not satisfied, the generated photons will destructively interfere with each other, limiting the number of generated photons that finally exit the crystal. However, the refractive index of an optical material is dependent on the wavelength of light propagating through the material, thus the phase matching condition might not be satisfied due to the relative phase delay among mixing photons, as shown in figure 4.

Figure 3: To enable efficient nonlinear wavelength conversion, both (a) photon energy conservation and (b) photon momentum conservation are to be satisfied simultaneously in a nonlinear wave mixing process.

Figure 4: Generated Photons vs. distance under phase-mismatching and phase-matching conditions.

For example, under plane wave and non-depleted pump assumption, wave mixing of input fundamental wave in a nonlinear dielectric medium with effective d-coefficient to generate output second harmonic wave, the generated output second harmonic intensity (power) vs. input fundamental intensity (power) can be derived as:

If a focused Gaussian beam is applied, taking SHG of a CW laser as an example, the frequency conversion efficiency can be deduced into a normalized efficiency as a function of wavelength. Please check "SHG normalized efficiency and pump wavelength dependence" for complete information.

**Birefrigence-Phase-Matching (BPM)**In order to achieve efficient wavelength conversion, phase matching among mixing waves is required. Traditionally, this has been done in nonlinear materials through Birefringence Phase Matching (BPM) technique, which orients crystal axis to a specific angle or adjust the crystal temperature to achieve phase matching condition among mixing waves.

In some limited conditions (as noncritical phase-matching), mixing waves are aligned such that they propagate along some axis of the birefrigent nonlinear crystal and the phase matching condition can be achieved by adjusting the crystal temperature. In many other conditions (as critical phase-matching), mixing waves are aligned at some angles to the axes of the index ellipsoid. Adjustment of the propagation direction to a specific selected phase-matching angle

*θ*or*φ*will change the refractive index of the extraordinary wave and thus achieve desired phase matching condition, as illustrated in figure 5.Figure 5: The refractive index of the fundamental wave must be equal to the refractive index of SHG wave to achieve birefringence phase matching. Therefore, the SHG wave will be oblique an angle θ (and/or

*φ)*to the optical axis (c-axis) of the nonlinear optical crystal.However, such a technique cannot enable efficient use of material's full transparency wavelength ranges at optimized conversion efficiency. In addition, BPM nonlinear wave mixing process occurs in the presence of both ordinary and extraordinary polarized radiations. When propagating along non-optical axis direction of a nonlinear optical crystal, the extraordinary wave suffers from birefringence walk-off; whereas ordinary ray will not. This walk-off issue (i.e. the converted wave diverges away from the fundamental wave) will distort the beam quality and also limit the effective interacting length (hence limiting the overall conversion efficiency).

**Quasi-Phase-Matching (QPM)**An alternative method to achieve phase matching condition among mixing waves is Quasi-Phase-Matching (QPM). Quasi-Phase-Matching (QPM) is a technique for engineering phase matching condition in nonlinear wave mixing process. In a QPM nonlinear wave mixing process, the phase mismatching is compensated by shifting the phase relative to each another among mixing waves over a coherent distance through inverting the sign of the nonlinear coefficient in nonlinear medium (such as “poling” to invert the crystal orientation for fabrication of Periodically Poled Lithium Niobate (PPLN)). By choosing the correct inversion period, the newly generated photons will effectively interfere constructively with previously generated photons, and as a result, the number of generated photons will grow as the light propagates through the QPM materials. The principle and effects of quasi-phase matching for nonlinear wave mixing process are illustrated in figure 6, where the wave vector associated with a periodic polarization modulation of a nonlinear medium is used to compensate the wave vector mismatching among mixing waves and thus to effectively achieve the phase matching condition.

Figure 6: (a) Photon energy conservation and (b) photon momentum conservation must be satisfied for efficient wave mixing process, where (c) the photon momentum conservation is achieved by creating a wave vector to compensate the wave vector mismatching among mixing waves.

As an example, figure 7 shows the effect of quasi-phase matching vs. non-phase-matching and phase-matching conditions for second-harmonic generation. In the nonlinear medium, the distance over which the accumulated phase difference of mixing waves reaches π is the "coherence length"

*L*_{c}.Figure 7: The growth of converted output power with distance for (i) phase-matching, (ii) non-phase-matching, and (iii) quasi-phase-mating interactions. The + & – sign represents the sign of the second-order nonlinear susceptibility.

**BPM (Birefringence-Phase-Matching) vs. QPM (Quasi-Phase-Matching)**Figure 8 summarizes the requirement for efficient nonlinear wave mixing process and also gives the comparison between BPM vs. QPM:

- BPM could give higher conversion efficiency for a few selected nonlinear wave mixing cases and does not require additional poling manufacture process. However, BPM is still required to satisfy phase-matching by tuning crystal temperature or crystal orientation, which is not achievable for all desired mixing wavelengths. In addition, BPM suffers from walk-off issues due to birefringence, which gives limitation for the beam quality and maximum interacting length for efficiency optimization.
- On the other hand, QPM gives flexibility and capability for engineering phase-matching conditions across all transparency ranges for selected nonlinear materials with maximum nonlinearity or with special nonlinearity (such as type II) for special applications. QPM also gives engineering flexibility for special spectrum engineering, which is beyond the reach by traditional BPM, thus virtually empower us to engineer a new type of material. However, QPM requires additional poling manufacture process which could be very challenging in several cases, such as poling manufacture process of a very short domain period material for UV range applications or poling manufacture process of a very large aperture material for high power applications. In addition, QPM poling manufacture process can not necessarily be realized in all nonlinear materials.

Figure 8: Summary and comparison of BPM vs. QPM for efficient nonlinear wave mixing.

**Note for practical consideration of using PPLN-QPM materials:**Above describes general QPM principle. However, in practical applications, different experimental setup may require different detailed considerations of using a PPLN crystal, which includes polarization, wavelength, temperature, and linewidth, etc.

**Polarization:**As mentioned, the input beam should be polarized at a certain direction to utilize the corresponding nonlinear coefficient. For example, type-0 need all e-polarized waves while o-polarized waves will be needed in type-I SHG.**Temperature/Wavelength:**Since the period form by periodical poling compensates only the design wavelength at a certain temperature, temperature/wavelength adjusting is needed for efficiency optimization. Example tuning curve (Normalized efficiency vs. wavelength) is shown in the figure 9 below.Figure 9: example tuning curve for 1560nm SHG.

**Linewidth:**In most cases, the PPLN is a highly dispersive material that the group velocity of the fundamental waves and converted waves are quite different, leading to a narrow acceptance bandwidth. The customer can check "Pump laser spectral width and wavelength conversion efficiency" to learn more about that.**Focusing/nornalized conversion efficiency vs. wavelength:**The frequency conversion is a nonlinear process, and normally need a high intensity. Focusing is normally needed for using a PPLN, and the optimal condition is L/b=2.84, where L is the crystal length and b is the confocal parameter and exactly twice of Raleigh range. Check "Focusing condition and wavelength conversion efficiency" and "SHG normalized efficiency and pump wavelength dependence" to see more about the focusing/nornalized conversion efficiency vs. wavelengths.**Short Pulse applications:**Due to material dispersion, different wavelength will have diffferent propagation speed (group velocity). Such an effect will become significant for short pulse (ps/fs) applications and limites the effective interactng lengths (thus overall efficiency and pulse spectrum). For more information about short pulse applications, please check "Broadband or pulsed SHG and temporal walk-off" and "Dispersion in PPLN".Please also contact us for any further technical supports of using PPLN-QPM materials.

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