Optical Three-Wave Coupling Processes
This chapter uses a reformed first-order frequency-domain wave equation for the isotopic medium to approximately describe the second-order nonlinear optics effects in the anisotropic medium. At first, the three-wave coupling equations are deduced, then ba
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Optical Three-Wave Coupling Processes
This chapter uses a reformed first-order frequency-domain wave equation for the isotopic medium to approximately describe the second-order nonlinear optics effects in the anisotropic medium. At first, the three-wave coupling equations are deduced, then based on these equations, several typical second-order nonlinear optics effects are studied: optical frequency doubling, sum frequency, difference frequency, and optical parameter amplification and parameter oscillation. The power conversion efficiency formulas for these effects are given. Finally, the basic concepts of phase matching are introduced based on the frequency doubling effect.
3.1 3.1.1
Three-Wave Coupled Equations Review of Second-Order Nonlinear Optics Effects in Isotopic Medium
Firstly we discuss the second-order nonlinear optics effect in general, it contains what specific effects, and we will give the polarizations of these effects in the isotopic material. Assuming that the incident light electrical fields consisted by two monochromatic light fields at the different frequencies and with same propagation direction, the total electrical field strength can be expressed as EðtÞ ¼
X
En eixn t þ c:c: ¼ E1 eix1 t þ E2 eix2 t þ c:c:
ð3:1:1Þ
n¼1;:2
In the isotopic medium without center symmetry, the second-order nonlinear polarization is
© Shanghai Jiao Tong University Press, Shanghai and Springer Nature Singapore Pte Ltd., 2017 C. Li, Nonlinear Optics, DOI 10.1007/978-981-10-1488-8_3
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Optical Three-Wave Coupling Processes
Pð2Þ ðtÞ ¼ e0 vð2Þ E2 ðtÞ:
ð3:1:2Þ
Substituting Eq. (3.1.1) into Eq. (3.1.2), after combination of the items with same frequency component, we obtain: Pð2Þ ðtÞ ¼ e0 vð2Þ ½ðE21 ei2x1 t þ E22 ei2x2 t þ 2E1 E2 eiðx1 þ x2 Þt þ 2E1 E2 eiðx1 x2 Þt Þ þ 2ðE1 E1 þ E2 E2 Þ þ c:c:
ð3:1:3Þ The Eq. (3.1.3) can be summarized by a simple formula, that is Pð2Þ ðtÞ ¼
X
Pð2Þ ðxi Þeixi t þ c:c;
ð3:1:4Þ
i
where i takes the positive integer. The polarization Pð2Þ ðxi Þ corresponds to the different second-order nonlinear optics effect with different susceptibility vð2Þ ðxi Þ; which is Pð2Þ ðxi Þ ¼ De0 vð2Þ ðxi ÞEðx1 ÞEðx2 Þ;
ð3:1:5Þ
where xi is the frequency of polarization field composed by two original monochromic fields at frequencies of x1 and x2 in different modes. Form Eq. (3.1.3) we can see that xi has five modes: 2x1 , 2x2 , x1 þ x2 , x1 x2 and 0. For second-order nonlinearity, n ¼ 2; the degeneration factor is D ¼ n!=m! ¼ 2=m!: When m ¼ 1; D = 2; when m = 2; D = 1. Therefore, corresponding to the different xi , the second-order nonlinear optics effects and corresponding polarizations are respectively: Optical frequency doubling Pð2x1 Þ ¼ e0 vð2Þ ð2x1 ÞE21 ;
ð3:1:6Þ
Optical frequency doubling Pð2x2 Þ ¼ e0 vð2Þ ð2x2 ÞE22 ;
ð3:1:7Þ
Pðx1 þ x2 Þ ¼ 2e0 vð2Þ ðx1 þ x2 ÞE1 E2 ;
ð3:1:8Þ
Optical sum frequency
Optical difference frequency Pðx1 x2 Þ ¼ 2e0 vð2Þ ðx1 x2 ÞE1 E2 ;
ð3:1:9Þ
3.1 Three-Wave Coupled Equations
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Optical rectification Pð0Þ ¼ 2e0 vð2Þ ð0ÞðE1 E1 þ E2 E2
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