Chapter VIII Second-Harmonic Generation and Parametric Oscillation Lecture 12 Chapter VIII Second-Harmonic Generation and Parametric Oscillation Highlights Nonlinear polarization 1. Nonlinear Optics Wave propagation in nonlinear media Phase-Matching 2. Second-Harmonic Generation SHG inside Resonator 3. Parametric Oscillation 4. Frequency Up-Conversion 参考书：《光电子学基础》，李家泽，阎吉祥著，北京理工大学出版社

§8.0 Introduction to Nonlinear Optics 非线性光学是专门研究强相干光与物质相互作用机理、现象和应用的光学分支。 光的电磁理论认为，光和物质相互作用的微观机理是光波电场引起介质极化，从而使介质中的原子（分子、离子）成为电偶极子，由于光波电场是时间的周期函数，所以各个偶极子将随时间作周期振动，这种受迫振动的频率与光波场的频率相同。并且这些振动的偶极子是一种场源，也要辐射电磁波，这就是通常所说的次波发射。 线性极化 线性光学： 非线性极化 非线性光学： i 阶非线性极化系数

§8.0 Introduction to Nonlinear Optics 以二阶极化为例： 单色平面电磁波： 极化波作为一个场源，不仅有基频光分量，还有直流分量和倍频分量。如果有多波作用，还会出现和频分量和差频分量。 需要指出的是，实际中一般并不容易同时观察到基频、倍频等光波，究竟哪一种频率的光波占主导作用，还要受到具体实验条件的限制。 1961年Franken的倍频实验标志着非线性光学的诞生。该领域的权威著作有 N. Bloembergen 的 “Nonlinear Optics”和 Y.R. Shen 的 “The Principles of Nonlinear Optics”。

§8.1 On The Physical Origin of Nonlinear Polarization x z (1) 线性振子模型和线性极化率

§8.1 On The Physical Origin of Nonlinear Polarization (2) 非线性振子模型 A. In a Symmetric Crystal No second order polarization exists B. In a Asymmetric Crystal Just consider the first two terms of restoring force, at D>0, a positive displacement (x>0) results in a larger force, than that of opposite direction. Indicate at the same E, electron moves shorter length in positive direction The above result means, at the same E, positive direction has smaller P

§8.1 On The Physical Origin of Nonlinear Polarization Optical E field Optical polarization Optical E field Polarization Fundamental frequency Double Frequency Steady DC Fourier Analysis

§8.1 On The Physical Origin of Nonlinear Polarization 单色平面电磁波： Assume the solution for x in the form First order: 与线性振子模型一致

§8.1 On The Physical Origin of Nonlinear Polarization Second order: 非线性光学系数 or

§8.1 On The Physical Origin of Nonlinear Polarization C. Nonlinear Optical Coefficients and Symmetry of Crystals Centro-symmetric crystals: No SHG occurs Noncentro-symmetric crystals: Depends on crystal’s conditions

§8.2 Formalism of Wave Propagation in Nonlinear Media 耦合波方程 (coupling wave functions) Consider three plane wave propagating along z direction with frequency

§8.2 Formalism of Wave Propagation in Nonlinear Media Nonlinear polarization contains: or These oscillate at the new frequencies, will not be able to drive the oscillation at , , or , unless Thus, we will have a power flow from the field and into that at

§8.2 Formalism of Wave Propagation in Nonlinear Media Assume: 振幅慢变化近似 耦合波方程

§8.3 Optical Second-Harmonic Generation (1) Conversion Efficiency Franken的倍频实验： No depleted input Output intensity is proportional to:

§8.3 Optical Second-Harmonic Generation Conversion efficiency of SHG To increase the conversion efficiency: Increase the fundamental frequency optical intensity b. Phase-matching factor gets maximum (2) Phase-Matching in SHG or Coherence length

§8.3 Optical Second-Harmonic Generation Coherence length is the maximum crystal length that is useful in producing the second-harmonic power Calculate the coherence length How to achieve phase-matching requirement In isotropic media  impossible In anisotropic media  same wave type also impossible Birefringence effect, different wave types

§8.3 Optical Second-Harmonic Generation Index ellipsoid Normal Surface s y o ne() x (no, 0, 0) (0, -no, 0) A B  (0, 0 , ne) y no ne no(q) ne(q) q Positive uniaxial crystal Negative uniaxial crystal If , is possible

§8.3 Optical Second-Harmonic Generation In a negative uniaxial crystal

§8.3 Optical Second-Harmonic Generation 第一类位相匹配 第二类位相匹配 匹配方式 位相匹配条件 晶体种类 正单轴 负单轴 Numerical Example: Second-Harmonic Generation KDP crystal: Calculate: Plot the index surface for illustration

§8.3 Optical Second-Harmonic Generation (3) Experimental Verification of Phase-Matching No phase-matching: Fig. 8-9 to see experimental results

§8.3 Optical Second-Harmonic Generation (4) SHG with Focused Gaussian Beams For

§8.3 Optical Second-Harmonic Generation For Example

§8.3 Optical Second-Harmonic Generation (5) SHG with a Depleted Input

§8.3 Optical Second-Harmonic Generation Manley-Rowe Relations Assume