# Introduction

Second-harmonic generation (SHG) is a case of nonlinear optical interaction; this phenomenon occurs when light (typically laser beam) goes through nonlinear material, generating a beam with higher energy. In other words, two photons of the same frequency generate a new photon with the double frequency of the input photons and shorter to half wavelength when they interact with nonlinear materials. SHG is also known as frequency doubling; this phenomenon was discovered by Franken and his colleagues about sixty years ago. (1) (2)

Frequency doubling occurs in a nonlinear material, crystals do not contain inversion centers, as determined by second-order nonlinear susceptibility .  Figure 1 a) illustrates geometry and b) illustrates energy-level of: (2) Figure 1. a) represent SHG. b) illustration of energy-level for SHG. (Figure: Miriam Yakob)

# Physical process

The insert beam generates a nonlinear polarization (P) wave interacting with nonlinear crystal because of the impact of  . The equation below describes nonlinear polarization P(t)   in a non-centrosymmetric material:

P(t)=   ϵ0χE(t)          (1)

Where: E is the strength of an optical field, below is general equation for field strength:

E(t) = Ee^-iωt+...      (2)

By combing equations 1 and 2 for second-order nonlinear polarization equation:

P^2(t)= 2ϵ0χ^2EE+(2ϵ0χ^2E^2e^(-i2ωt)+...)             (3)

ϵ0 is the permittivity of free space, and t is for time. (2) (3)

The nonlinear polarization wave scatters higher energy beam with twice the frequency in the same direction of the input beam, generating two fields: electromagnetic, and due to phase-matching, second harmonic. Second-harmonic field interacts also with the inserted beam and results energy moving from input photons to the second harmonic photons. (3)

# Phase matching

Phase matching in SHG-material means that the vectors of input wave and output second-harmonic wave are the same (𝞓k= 0), thus optical indices n(ω)= n(2ω). Phase matching is essential for SHG uses in different applications; even a small amount of mismatching has a huge negative impact on SHG efficiency. (3) (4) (9) There are two types of phase-matching: Critical or angle phase matching and noncritical. In the first one, the wave is either refracts at an angle (extraordinary polarization) or goes straight through the material (ordinary polarization) (5)(6)(1). In the latter, phase matching is achieved by modifying the temperature of the material (7).

# SHG Materials

As mentioned before, SHG occurs in non-centrosymmetric materials meaning materials without inversion centers. For high performance in different applications, nonlinear crystal should include phase matching and large band gap, additionally to transparency. (8) some materials used for SHG:

• borate crystals such as bismuth borate (BiB3O6), cesium lithium borate (CsLiB6O10), and β phase barium borate (β-BaB2O4). Figure 2 shows the crystal structure of a) bismuth borate b) cesium lithium borate c) (β-BaB2O4). (1‎) (10)

a                                                           b                                                             c Figure 2. Crystal structures of a) BiB3O6 b) CsLiB6O10 c) β-BaB2O4. (Figure: Miriam Yakob)

• Niobate crystals like lithium niobate (LiNbO3) and potassium niobate (KNbO3) (fig 3). (‎1) (‎10)

a                                                                        b Figure 3. 3D crystal structure a) LiNbO3 b) KNbO3. (Figure: Miriam Yakob)

• Other materials fabricated and used for SHG applications: monopotassium phosphate (KH2PO4) (fig 4 a) (8), and α- and β-silver polyiodates (α-AgI3O8 and β-AgI3O8) (fig 4 b).

a                                                                  b Figure 4. Structures of a) AgI3O8.  b) KH2PO4  (Figure: Miriam Yakob)

# Applications

SHG is used in laser technology to make specific types of lasers (‎8). SHG phenomena are utilized in microscope imaging for biomedical applications; SHG imaging microscopy is used for disease discovery (‎11).

# References

1. Second-harmonic generation. 2.2022 Wikipedia. Referred 14.03.2022.
2. Boyd. (2007) “The Nonlinear Optical Susceptibility, book Elsevier”. Nonlinear Optics, book (third ed.) 1-67.
3. Paschotta. N. d. Frequency doubling; RP Photonics Encyclopedia. Retrieved 14.3.2022. Website.
4. Zhang et al., Phase-Matching in Nonlinear Optical Compounds: A Materials Perspective; Chemistry of Materials 2017 (7) 2655-2668.
5. Paschotta. N. d. Critical phase matching; RP Photonics Encyclopedia. Retrieved 15.03.20212. Website.
6. Double Refraction. N. d. Britannica. Retrieved 15.03.2022. Website.

https://www.britannica.com/science/double-refraction

1. Paschotta, n. d. Noncritical Phase Matching; RP Photonics Encyclopedia. Retrieved 15.03.2022.
2. Chen et al., High-Performance Second Harmonic-Generation (SHG) Materials: New Developments and New Strategies; American Chemical Society 2021 (12) 2775-2783.

https://doi.org/10.1021/acs.accounts.1c00188

1. Eckardt et al., Phase Matching Limitations of High Efficiency Second-Harmonic Generation; IEEE Journal of Quantum Electronics 1984 (20) 1178-1187.
2. Pascotta, n. d. Nonlinear Crystal Materials; RP Photonics Encyclopedia. Retrieved 16.03.2022.
3. Campagnola, Second Harmonic Generation Imaging Microscopy: Applications to Diseases Diagnostics; Analytical chemistry 2011 (83) 3224-3231. (https://doi.org/10.1021/ac1032325)

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