Second harmonic generation

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Second harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process, in which photons interacting with a nonlinear material are effectively "combined" to form new photons with twice the energy, and therefore twice the frequency and half the wavelength of the initial photons.

Second harmonic generation was first demonstrated by P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich at the University of Michigan, Ann Arbor, in 1961. The demonstration was made possible by the invention of the laser, which created the required high intensity monochromatic light. They focused a ruby laser with a wavelength of 694 nm into a quartz sample. They sent the output light through a spectrometer, recording the spectrum on photographic paper, which indicated the production of light at 347 nm. Famously, when published in the journal Physical Review Letters, the copy-editor mistook the dim spot (at 347 nm) on the photographic paper as a speck of dirt and removed it from the publication.^{[citation needed]}

1 Derivation of Second Harmonic Generation
2 Second Harmonic Generation with Depletion
3 Types of SHG
4 References

[edit] Derivation of Second Harmonic Generation

The simplest case for analysis of second harmonic generation is a plane wave of amplitude $E (ω)$ traveling in a nonlinear medium in the direction of its $k$ vector. A polarization is generated at the second harmonic frequency

$P(2\omega) = 2\epsilon_0d_{eff}(2\omega ;\omega,\omega)E^2(\omega), \,$

where $2 d e f f = χ (2)$ .The wave equation at $2ω$ (assuming negligable loss and asserting the slowly varying envelope approximation) is

$\frac{\partial E(2\omega)}{\partial z}=-\frac{i\omega}{n_{2\omega}c}d_{eff}E^2(\omega)e^{i\Delta k z}$

where $Δ k = k (2ω) - 2 k (ω)$ .

At low conversion efficiency ( $E (2ω) < < E (ω)$ the amplitude $E (ω)$ remains essentially constant over the interaction length, $l$ . Then, with the boundary condition $E (2ω, z = 0) = 0$ we get

$E(2\omega,z=l)=-\frac{i\omega}{n_{2\omega}c}E^2(\omega)\int_0^l{e^{i\Delta k z}}=-\frac{i\omega}{n_{2\omega}c}E^2(\omega)l\frac{\sin{\Delta k l/2}}{\Delta k l/2}e^{i\Delta k l/2}$

In terms of the optical intensity, $I=n/2\sqrt{\epsilon_0/\mu_0}|E|^2$ , this is,

$I(2\omega,l)=\frac{2\omega^2d^2_{eff}l^2}{n_{2\omega}n_{\omega}^2c^3\epsilon_0}(\frac{\sin{(\Delta k l/2)}}{\Delta k l/2})^2I^2(\omega)$

This intensity is maximized for the phase matched condition $Δ k = 0$ . If the process is not phase matched, the driving polarization at $2ω$ goes in and out of phase with generated wave $E (2ω)$ and conversion oscillates as $sin(Δ k l / 2)$ . The coherence length is defined as $l_c=\frac{\pi}{\Delta k}$ . It does not pay to use a nonlinear crystal much longer than the coherence length. (Periodic poling and Quasi-phase-matching provide another approach to this problem.)

[edit] Second Harmonic Generation with Depletion

When the conversion to second harmonic becomes significant it becomes necessary to include depletion of the fundamental. One then has the coupled equations:

$\frac{\partial E(2\omega)}{\partial z}=-\frac{i\omega}{n_{2\omega}c}d_{eff}E^2(\omega)e^{i\Delta k z}$ ,

$\frac{\partial E(\omega)}{\partial z}=-\frac{i\omega}{n_{\omega}c}d_{eff}^*E(2\omega)E^*(\omega)e^{-i\Delta k z}$ ,

where $*$ denotes the complex conjugate. For simplicity, assume phase matched generation ( $Δ k = 0$ ). Then, energy conservation requires that

$n_{2\omega}[E^*(2\omega)\frac{\partial E(2\omega)}{\partial z}+c.c.]=-n_\omega[E(\omega)\frac{\partial E^*(\omega)}{\partial z}+c.c.]$

$n_{2\omega}|E(2\omega)|^2+n_\omega|E(\omega)|^2=n_{2\omega}E_0^2$ .

Now we solve the equations with the premise

$E (ω) = | E (ω) | e i φ(ω)$

$E (2ω) = | E (2ω) | e i φ(2ω)$

We get

$\frac{d|E(2\omega)|}{dz}=-\frac{i\omega d_{eff}}{n_\omega c}[E_0^2-|E(2\omega)|^2]e^{2i\phi(\omega)-i\phi(2\omega)}$

$\int_0^{|E(2\omega)|l}{\frac{d|E(2\omega)}{E_0^2-|E(2\omega)|^2}}=-\int_0^l{\frac{i\omega d_{eff}}{n_\omega c}dz}$

Using

$\int{\frac{dx}{a^2-x^2}}=\frac{1}{a}\tanh^-1{\frac{x}{a}}$

we get

$|E(2\omega)|_{z=l}=E_0\tanh{(\frac{-iE_0l\omega d_{eff}}{n_\omega c}e^{2i\phi(\omega)-i\phi(2\omega)})}$

If we assume a real $d e f f$ , the relative phases for real harmonic growth must be such that $e 2 i φ(ω) - i φ(2ω) = i$ . Then

$I(2\omega,l)=I(\omega,0)\tanh^2(\frac{E_0\omega d_{eff}l}{n_\omega c})$

$I (2ω, l) = i (ω,0) t a n h 2 (Γ l)$ ,

where $Γ = ω d e f f E 0 / n c$ . From $I (2ω, l) + I (ω, l) = I (ω,0)$ , it also follows that

$I (ω, l) = I (ω,0) s e c h 2 (Γ l)$ .

[edit] Types of SHG

Second harmonic generation occurs in two types, denoted I and II. In Type I SHG two photons having ordinary polarization with respect to the crystal will combine to form one photon with double the frequency and extraordinary polarization. In Type II SHG, two photons having orthogonal polarization will combine to form one photon with double the frequency and extraordinary polarization. For a given crystal orientation, only one of these type of SHG occurs.

[edit] References

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 7, p. 118–119 (1961). DOI: DOI:10.1103/PhysRevLett.7.118
K. R. Parameswaran, J. R. Kurz, R. V. Roussev, M. M. Fejer, "Observation of 99% pump depletion in single-pass second-harmonic generation in a periodically poled lithium niobate waveguide", Optics Letters, 27, p. 43-45 (January 2002).
Frequency doubling. Encyclopedia of laser physics and technology. Retrieved on 2006-11-04.