Second-harmonic generation

Diagram of the second harmonic generation process.

Second harmonic generation (also called frequency doubling or abbreviated SHG) 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. It is a special case of sum frequency generation.

Second harmonic generation was first demonstrated by Peter 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.

Types of SHG

Second harmonic generation occurs in three types, denoted 0, I and II. In Type 0 SHG two photons having extraordinary polarization with respect to the crystal will combine to form a single photon with double the frequency/energy and extraordinary polarization. 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 polarizations will combine to form one photon with double the frequency and extraordinary polarization. For a given crystal orientation, only one of these types of SHG occurs. In general to utilise Type 0 interactions a Quasi-phase-matching crystal type will be required, for example periodically poled lithium niobate (PPLN).

Second harmonic generation microscopy

Main article: Second harmonic imaging microscopy

In biological and medical science, the effect of second harmonic generation is used for high-resolution optical microscopy. Because of the non-zero second harmonic coefficient, only non-centrosymmetric structures are capable of emitting SHG light. One such structure is collagen, which is found in most load-bearing tissues. Using a short-pulse laser such as a femtosecond laser and a set of appropriate filters the excitation light can be easily separated from the emitted, frequency-doubled SHG signal. This allows for very high axial and lateral resolution comparable to that of Confocal microscopy without having to use pinholes. SHG microscopy has been used for extensive studies of the Cornea and Lamina cribrosa sclerae, both of which consist primarily of collagen.

Other uses

Second harmonic generation is used by the laser industry to make green 532 nm lasers from an 808 nm source. The source is used to pump a Nd:YAG crystal laser to generate light at 1064 nm. This 1064 nm light is fed through a bulk KDP crystal. In high-quality diode lasers the crystal is coated on the output side with an infrared filter to prevent leakage of intense 1064 nm or 808 nm infrared light into the beam. Both of these wavelengths are invisible and do not trigger the defensive "blink-reflex" reaction in the eye and can therefore be a special hazard to the human eyes. Furthermore, some laser safety eyewear intended for argon or other green lasers may filter out the green component (giving a false sense of safety), but transmit the infrared. Nevertheless, some "green laser pointer" products have become available on the market which omit the expensive infrared filter, often without warning.

Historical note

Generating the second harmonic, often called frequency doubling, is also a process in radio communication; it was developed early in the 20th century, and has been used with frequencies in the megahertz range.

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

$P(2\omega )=2\epsilon _{0}d_{\text{eff}}(2\omega ;\omega ,\omega )E^{2}(\omega ),\,$

The wave equation at 2ω (assuming negligible loss and asserting the slowly varying envelope approximation) is

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

where $\Delta k=k(2\omega )-2k(\omega )$.

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

$E(2\omega ,z=l)=-{\frac {i\omega d_{\text{eff}}}{n_{2\omega }c}}E^{2}(\omega )\int _{0}^{l}{e^{i\Delta kz}dz}=-{\frac {i\omega d_{\text{eff}}}{n_{2\omega }c}}E^{2}(\omega )l\,{\frac {\sin {(\Delta kl/2)}}{\Delta kl/2}}e^{i\Delta kl/2}$

In terms of the optical intensity, $I=n/2{\sqrt {\epsilon _{0}/\mu _{0}}}|E|^{2}$, this is,

$I(2\omega ,l)={\frac {2\omega ^{2}d_{\text{eff}}^{2}l^{2}}{n_{2\omega }n_{\omega }^{2}c^{3}\epsilon _{0}}}\left({\frac {\sin {(\Delta kl/2)}}{\Delta kl/2}}\right)^{2}I^{2}(\omega )$

This intensity is maximized for the phase matched condition Δk = 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(Δkl/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.)

Second harmonic generation with depletion

When the conversion to 2nd harmonic becomes significant it becomes necessary to include depletion of the fundamental. One then has the coupled equations:

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

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

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

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

where $c.c.$ is the complex conjugate of the other term, or

$n_{2\omega }|E(2\omega )|^{2}+n_{\omega }|E(\omega )|^{2}=n_{2\omega }E_{0}^{2}$.

Now we solve the equations with the premise

$E(\omega )=|E(\omega )|e^{i\phi (\omega )}$

$E(2\omega )=|E(2\omega )|e^{i\phi (2\omega )}$

and obtain

${\frac {d|E(2\omega )|}{dz}}=-{\frac {i\omega d_{\text{eff}}}{n_{\omega }c}}\left[E_{0}^{2}-|E(2\omega )|^{2}\right]e^{2i\phi (\omega )-i\phi (2\omega )}$

$\int _{0}^{|E(2\omega )|l}{\frac {d|E(2\omega )|}{E_{0}^{2}-|E(2\omega )|^{2}}}=-\int _{0}^{l}{{\frac {i\omega d_{\text{eff}}}{n_{\omega }c}}dz}$

Using

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

we get

$|E(2\omega )|_{z=l}=E_{0}\tanh {\left({\frac {-iE_{0}l\omega d_{\text{eff}}}{n_{\omega }c}}e^{2i\phi (\omega )-i\phi (2\omega )}\right)}$

If we assume a real $d_{\text{eff}}$, the relative phases for real harmonic growth must be such that $e^{2i\phi (\omega )-i\phi (2\omega )}=i$. Then

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

or

$I(2\omega ,l)=I(\omega ,0)\tanh ^{2}{(\Gamma l)},$

where $\Gamma =\omega d_{\text{eff}}E_{0}/nc$. From $I(2\omega ,l)+I(\omega ,l)=I(\omega ,0)$, it also follows that

$I(\omega ,l)=I(\omega ,0)\mathrm {sech} ^{2}{(\Gamma l)}.$

External links

Articles on second harmonic generation

• Parameswaran, K. R., Kurz, J. R., 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 2006-11-04.