Second-harmonic Generation

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'' (citation below), 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.

In recent years, SHG has been extended to biological applications. Researchers Leslie Loew and Paul Campagnola at the University Of Connecticut have applied SHG to imaging of molecules that are intrinsically second-harmonic-active in live cells, such as collagen, while Joshua Salafsky Biodesy is pioneering the technique's use for studying biological molecules by labeling them with second-harmonic-active tags, in particular as a means to detect conformational change at any site and in real time. SH-active unnatural amino acids can also be used as probes.

DERIVATION OF SECOND HARMONIC GENERATION

The simplest case for analysis of second harmonic generation is a plane wave of amplitude

E(\omega)

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

2d_{eff}=\chi^{(2)}

.The wave equation at

2\omega

(assuming negligible loss and asserting the slowly varying envelope approximation) is

:

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

where

\Delta k=k(2\omega)-2k(\omega)

.

At low conversion efficiency (

E(2\omega)< the amplitude E(\omega) remains essentially constant over the interaction length, l . Then, with the boundary condition E(2\omega,z=0)=0 we get

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

This intensity is maximized for the phase matched condition

\Delta k=0

. If the process is not phase matched, the driving polarization at

2\omega

goes in and out of phase with generated wave

E(2\omega)

and conversion oscillates as

\sin{(\Delta k l/2)}

. The coherence length is defined as

l_c=rac{\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.)

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:

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

E(2\omega)E^---(\omega)e^{-i\Delta k z},

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

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

where

c.c.

is the complex conjugate of the other term, or

<math>E(\omega) E(\omega)e^{i\phi(\omega)}</math>
<math>E(2\omega) E(2\omega)e^{i\phi(2\omega)}</math>

we get

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Second-harmonic Generation