Interactive length of fundamental wave and second harmonic generated on the surface of anomalous dispersion medium

Xiaojing Wang; Huaijin Ren; Gang Wang; Jun He

doi:10.3788/COL201917.081902

Journals >Chinese Optics Letters >Volume 17 >Issue 8 >Page 081902 > Article

Chinese Optics Letters
Vol. 17, Issue 8, 081902 (2019)

Interactive length of fundamental wave and second harmonic generated on the surface of anomalous dispersion medium

Xiaojing Wang¹, Huaijin Ren², Gang Wang¹, and Jun He^1,*

Author Affiliations

¹Institute of Super-Microstructure and Ultrafast Process in Advanced Materials, School of Physics and Electronics, Central South University, Changsha 410083, China

²Institute of Applied Electronics, China Academy of Engineering Physics, Mianyang 621900, China

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DOI: 10.3788/COL201917.081902 Cite this Article Set citation alerts

Xiaojing Wang, Huaijin Ren, Gang Wang, Jun He, "Interactive length of fundamental wave and second harmonic generated on the surface of anomalous dispersion medium," Chin. Opt. Lett. 17, 081902 (2019) Copy Citation Text

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Abstract

In this Letter, a new method is presented to calculate the interactive length between the fundamental wave and second harmonic generation (SHG) for the configuration of total internal reflection on the inner surface of a nonlinear crystal. Three independent experiments are designed to measure the bandwidths of this second harmonic wave. The theoretical expression of the intensity of SHG is obtained through a nonlinear coupled wave equation. The interactive length of this phase-matched SHG can be calculated mathematically by utilizing the measured bandwidths and the intensity equation. There is no existing method to obtain the interactive length either from theoretical calculations or by experimental measurement. This method can be applied to estimate the extremely short interactive volume in nonlinear processes.

E_{1} = A_{1} e^{- z^{2} / a^{2}} e^{i (k_{1} \cos θ \cdot x + k_{1} \sin θ \cdot z)},

(1)

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E_{1}^{'} = A_{1} e^{- z^{2} / a^{2}} e^{i (k_{1} \cos θ \cdot x - k_{1} \sin θ \cdot z)} .

(2)

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E_{2} = A_{2} (x, z) e^{i k_{2} \cdot x} .

(3)

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\frac{\partial}{\partial x} A_{1} (x, z) = 0,

(4)

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(\frac{i}{2 k_{2}} \frac{\partial^{2}}{\partial z^{2}} + \frac{\partial}{\partial x}) A_{2} (x, z) = - i \frac{k_{2}}{2 n_{2 ω}^{2}} h (z) χ^{(2)} {(A_{1} e^{- \frac{z^{2}}{a^{2}}})}^{2} \cdot e^{i (k_{2} - 2 k_{1} \cos θ) \cdot x},

(5)

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h (z) = {\frac{1,}{0}} \frac{z \geq 0}{z < 0} .

(6)

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A_{2} (x, k_{z}) = \int A_{2} (x, z) \cdot e^{i k_{z} z} d z .

(7)

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(\frac{\partial}{\partial x} - i \frac{k_{z}^{2}}{2 k_{2}}) A_{2} (x, k_{z}) = - i \frac{k_{2}}{2 n_{2 ω}^{2}} χ^{(2)} {(A_{1})}^{2} e^{i Δ k x} \cdot \int e^{- 2 z^{2} / a^{2}} \cdot h (z) \cdot e^{i k_{z} z} d z .

(8)

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A_{2} (k_{z}, x) = i \frac{k_{2}}{2 n_{2 ω}^{2}} χ^{(2)} \cdot A_{1}^{2} \cdot \frac{e^{i (Δ k - \frac{k_{z}^{2}}{2 k_{2}}) x} - 1}{i (Δ k - \frac{k_{z}^{2}}{2 k_{2}})} \cdot e^{i \frac{k_{z}^{2}}{2 k_{2}} x} \cdot \int e^{- 2 z^{2} / a^{2}} \cdot h (z) \cdot e^{i k_{z} z} d z .

(9)

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I_{2} (k_{z}, x) = {[\frac{k_{2}}{2 n_{2 ω}^{2}} χ^{(2)}]}^{2} I_{1}^{2} \cdot x^{2} \cdot {\sin c}^{2} [(Δ k - \frac{k_{z}^{2}}{2 k_{2}}) \frac{x}{2}] {| H (k_{z}) |}^{2} .

(10)

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I_{2} (k_{z}, x) = {[\frac{k_{2}}{2 n_{2 ω}^{2}} χ^{(2)}]}^{2} I_{1}^{2} \cdot x^{2} \cdot {sinc}^{2} | Δ k \cdot \frac{x}{2} | {| H (k_{z}) |}^{2},

(11)

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Δ k = 4 π [n_{2 e} (T) - n_{1 o} (T) \cdot \cos θ] / λ_{1},

(12)

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Δ k \cdot x / 2 = 1.3916 .

(13)

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Xiaojing Wang, Huaijin Ren, Gang Wang, Jun He, "Interactive length of fundamental wave and second harmonic generated on the surface of anomalous dispersion medium," Chin. Opt. Lett. 17, 081902 (2019)

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