**PPLN Guide: Overview****Principle: Introduction of nonlinear wavelength conversion****Light-matter interaction****Phase-matching****Birefringence-Phase-Matching (BPM)****Quasi-Phase-Matching (QPM)****BPM vs. QPM**

**Materials: Material properties of PPLN****Transmission****Refractive index****Strength of nonlinearity & Phase-matching configuration****Damage mechanism**

**Applications: QPM configurations & applications****Efficient wavelength conversion across full transparency range****Design and realization of different QPM patterns****Conversion efficiency enhancement via QPM waveguides****Beyond**

**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 the electric field (**E**) and generate induced dielectric polarization (**P**). The induced dielectric polarization (**P**) describes the overall response behavior of a dielectric medium to the electric field (**E**). In a linear dielectric medium, the response is linear (as shown in figure 1-a) and the frequency of light will not be altered. On the other hand, in the nonlinear dielectric medium, the response is nonlinear (as shown in figure 1-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-a (left): Linear dielectric polarization. Figure 1-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}).**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 2. 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 3.

Figure 2: 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 3: Generated Photons vs. distance under phase-mismatching and phase-matching conditions.

**Birefringence-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 adjusts 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 4.Figure 4: 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 5, 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.

Fig.ure 5: (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 6 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 6: 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.

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

**Principle: Introduction of nonlinear wavelength conversion****Light-matter interaction****Phase-matching****Birefringence-Phase-Matching (BPM)****Quasi-Phase-Matching (QPM)****BPM vs. QPM**

**Materials: Material properties of PPLN****Transmission****Refractive index****Strength of nonlinearity & Phase-matching configuration****Damage mechanism**

**Applications: QPM configurations & applications****Efficient wavelength conversion across full transparency range****Design and realization of different QPM patterns****Conversion efficiency enhancement via QPM waveguides****Beyond**

**Materials: Material properties of PPLN**From practical application consideration of nonlinear wave mixing process, the key properties of nonlinear optical materials can be characterized by (1) transparency range, (2) refractive index, (3) the strength of nonlinearity and phase-matching scheme, as well as (4) damage threshold. The transparency range ensures the nonlinear wave mixing process is operated at no-absorption range. The material refractive index is related to phase matching design and mixing efficiency. The strength of nonlinearity and phase-matching scheme is the dominating factor for the wave mixing efficiency optimization of the selected nonlinear optical material. The damage threshold characterizes the maximum input/output power that the device can handle without inducing laser damage on surface or inside the material. Summary of transparency range and other key factors for selecting nonlinear materials are shown in table 1.

Table 1: Summary of key nonlinear material properties for efficient nonlinear wave mixing

**Transmission**An example transmission spectrum is shown in figure 1, which gives transmission in % vs. wavelength for uncoated congruent LiNbO

_{3}(CLN). The transmission in % is not shown to be 100% between the UV and MIR transparency range due to the reflection of non-AR coated surfaces, which can be corrected with proper AR coating. However, the steep slope at near UV and near MIR range is due to material absorption. In addition, there is a transmission dip near 2870 nm due to OH- absorption.Figure 1: Transmission curves of CLN

By looking into the transmission edges, figure 2 gives a transmission in percentage as a function of the wavelength in ordinary and extraordinary waves at near-UV and near-MIR region for uncoated materials of congruent lithium tantalate (CLT), 7 mol.% doped MgO doped CLT and 5 mol %. doped CLN. CLT material family has wider transparency range (especially near UV range) as compared to that of CLN, thus one would choose the CLT material family if the application wavelength is beyond the transparency range of CLN.

Figure 2: Transmission spectra of 1-mm-thick CLT, Mg7% -doped CLT, and Mg5% doped CLN in (a) UV region for ordinary wave, (b) UV region for extraordinary wave, (c) MIR region for ordinary wave, and (d) MIR region for the extraordinary wave. (Reference from: “Mg-doped congruent LiTaO3 crystal for large-aperture quasi-phase matching device”, Hideki Ishizuki and Takunori Taira, Vol. 16, Issue 21, pp. 16963-16970 (2008)).

Although typically the transmission begins to drop when the wavelength <400 nm and >4000 nm, the CLN is still applicable in those ranges but with a higher absorption loss.

**Refractive index**The temperature dependent refractive index is described by the Sellmeier equation below

where the temperature-dependent parameter

*f*isand the Sellmeier coefficients are

This property describes the chromatic dispersion (refractive index vs. wavelength) and birefringence properties (polarization dependence) and thus determines the phase-matching condition, phase-matching bandwidth, etc.

**Strength of nonlinearity & Phase-matching configuration**Second order nonlinearity characterizes the strength of response for a dielectric medium to electric field. For convenience, we can define a two dimensional (6x3) nonlinear dielectric tensor, commonly known as Kleinman

**d**-tensor, to describe the strength of nonlinearity:In practical situation, depending on the crystal class, many tensor components actually can be reduced to zero for crystal structure symmetry reason. For example, the

**d**-tensor of LiNbO_{3}(which is triangular 3m crystal class) can be expressed asWhere the nonlinear d coefficient is

*d*_{22}=2.59pm/V,*d*_{31}=4.85pm/V,*d*_{33}=25.3pm/V. Magnitude of the effective nonlinear coefficient*d*_{eff}with units of pm/V describes the nonlinear frequency mixing efficiency. For QPM configuration with polarization along z-axis, the effective*d*_{eff}will be reduced by a 2/π factor, i.e.*d*_{eff}=(2/π)*d*_{33}.**Damage mechanism**Damage threshold describes the highest power that a nonlinear material can sustain without causing surface or material damage or other undesired effects. The mechanism for laser induced damage threshold (LIDT) could be due to (a) thermal effect, (b) laser induced breakdown, and (c) surface damage etc. Additional undesired effect includes nonlinear self-focusing, photo-darkening (gray tracking), and GRIIA (green-induced infrared absorption) etc.

From practical application consideration, the laser induced damage threshold (LIDT) will depend on the engineering optimization of operation wavelength, peak/averaged power, spatial focusing condition, temporal pulse width, pulse energy, surface quality, and also integration packaging, etc.

**Principle: Introduction of nonlinear wavelength conversion****Light-matter interaction****Phase-matching****Birefringence-Phase-Matching (BPM)****Quasi-Phase-Matching (QPM)****BPM vs. QPM**

**Materials: Material properties of PPLN****Transmission****Refractive index****Strength of nonlinearity & Phase-matching configuration****Damage mechanism**

**Applications: QPM configurations & applications****Efficient wavelength conversion across full transparency range****Design and realization of different QPM patterns****Conversion efficiency enhancement via QPM waveguides****Beyond**

**Applications: QPM configurations & applications**With its unique capabilities for engineering phase-matching conditions within nonlinear materials, Quasi-Phase Matching (QPM) allows a variety of applications beyond what can be realized by traditional Birefringence-Phase-Matching (BPM). For examples, QPM enables

- Efficient wavelength conversion across full transparency range of selected nonlinear materials under a variety of wave mixing configurations.

- Design and realization of different QPM patterns for diverse applications.

- Conversion efficiency enhancement via QPM based waveguides.

**Efficient wavelength conversion across full transparency range of selected nonlinear materials under a variety of wave mixing configurations**As a difference from birefringence phase matching (BPM), QPM technique can possibly utilize full transparency ranges of the selected non-linear materials (350 nm~5000 nm for PPLN) and also access to the highest nonlinear d-coefficient (such as for PPLN) to achieve efficient wavelength conversion. Since spatial walk-off issue in general will not occur in QPM configuration, if needed, one can also use the maximum available device length for efficiency optimization (50mm long is common and up to 80mm long is also available for PPLN). This enables efficient wavelength conversion via different wave-mixing configurations for application optimization. Examples of wave mixing configurations (SHG/SFG/DFG/OPA/OPG/OPO) are illustrated in figure 1:

Figure 1: Illustrate wave mixing configuration of (a) SHG, (b) SFG, (c) DFG, (d) OPA, (e) OPG, and (f) OPO.

By designing proper phase matching period, QPM technique will also allow specific type of wave mixing such as Type 0, Type I and Type II for user’s specific application requirement, as illustrated in figure 2. With QPM, all above types can possibly be realized for selected QPM materials, as one can engineer the phase-matching condition among mixing waves:

(a) In Type 0, two photons having extraordinary polarizations will combine to form a new photon with extraordinary polarization.

(b) In Type I, two photons having ordinary polarizations will combine to form a new photon with extraordinary polarization.

(c) In Type II, two photons having orthogonal polarizations will combine to form a new photon with ordinary polarization.

Figure 2: Illustrate wave mixing of (a) Type 0, (b) Type I and (c) Type II.

From practical consideration, there are several basic configurations for efficiency optimization:

**Single-pass bulk configuration****Single-pass waveguide configuration****Cavity (such as OPO) configuration****Pulse (by ns or ns/ps pulse) configuration**

**Design and realization of different QPM patterns for diverse applications.**Other advantages of QPM technology are that QPM domain structures are defined by photo-mask patterning process, which provides additional application dimensions unavailable by traditional nonlinear bulk materials. Several other special periodic pattern designs, such as cascaded, fan-out, and other customized configurations are also available for specific applications using the CW, ns pulsed and ultra-short ps/fs pulsed laser pumps. Examples of special QPM structures are illustrated in figure 3

Figure 3: Example of various QPM patterns: Uniform, Multiple (Tunable), Cascaded (SHG+SFG/DFG), Chirped (Spectrum Engineering), Fan-out (Continuous tunable) etc.

**Conversion efficiency enhancement via QPM based waveguides**With the capabilities for engineering phase-matching condition along the desired crystal orientation, QPM also allows the feasibility for fabricating the waveguide in the selected periodically poled materials, which enhance the device efficiency by 2 or 3 orders of magnitude larger than that of the bulk materials and thus enable several applications beyond what can be achieved by the same type of bulk materials. While QPM bulk device offers advantages of higher power and larger aperture applications, QPM waveguide (WG) can be used to further enhance nonlinear wave mixing efficiency by tightly confining the laser intensity over long propagation distance, as shown in figure 4.

Figure 4: Comparison of QPM Bulk device and QPM waveguide device

**Beyond**

With its unique capabilities for engineering phase-matching condition within nonlinear materials, Quasi-Phase Matching (QPM) enables the possibilities to access photon’s full dimensions (wavelength dimension from UV to mid-IR/THz, time dimension from CW to ps/fs/as, and energy dimension from single photon to very huge photon numbers (Watts & up)) for selected nonlinear materials (such as PPLN). As illustrated in figure 5, such a full-spectrum capability of PPLN, along with the advance of other complementary photonics technologies (DPSS-diode pump solid state lasers and fiber lasers; sensor and spectrometer; other wavelength management technologies, etc.), will make several critical contributions to a variety of important applications.

Figure 5: small photon’s (a) full spectrum dimensions and (b) potential BIG impacts.

**Principle: Introduction of nonlinear wavelength conversion****Light-matter interaction****Phase-matching****Birefringence-Phase-Matching (BPM)****Quasi-Phase-Matching (QPM)****BPM vs. QPM**

**Materials: Material properties of PPLN****Transmission****Refractive index****Strength of nonlinearity & Phase-matching configuration****Damage mechanism**

**Applications: QPM configurations & applications****Efficient wavelength conversion across full transparency range****Design and realization of different QPM patterns****Conversion efficiency enhancement via QPM waveguides****Beyond**

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