the amplitude remains essentially constant over the interaction length, . Then, with the boundary condition we getIn terms of the optical intensity, , this is,
This intensity is maximized for the phase matched condition . If the process is not phase matched, the driving polarization at goes in and out of phase with generated wave and conversion oscillates as . The coherence length is defined as . 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.)
econd 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:
,
,
where denotes the complex conjugate. For simplicity, assume phase matched generation (). Then, energy conservation requires that
where is the complex conjugate of the other term, or
.
Now we solve the equations with the premise
We get
Using
we get
If we assume a real , the relative phases for real harmonic growth must be such that . Then
or
,
where . From , it also follows that
.
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.
Common Uses
Second harmonic generation is used by the laser enthusiast industry to make green 532nm lasers from an 808nm source. The source is converted to 1064nm by a crystal, then fed through a KDP second harmonic crystal. This is capped by an infrared filter to prevent leakage of any infrared that would be harmful to the human eyes
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 MHz range.
External links
Notes
Articles on Second Harmonic Generation
*Franken, P. A., Hill, A. E., Peters, C.W., and Weinreich, G., " [http://link.aps.org/abstract/PRL/v7/p118 Generation of Optical Harmonics] ", "Phys. Rev. Lett." 7, p. 118–119 (1961). DOI: doi|10.1103/PhysRevLett.7.118
*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).
*cite web|url=http://www.rp-photonics.com/frequency_doubling.html |title=Frequency doubling|work= Encyclopedia of laser physics and technology|accessdate=2006-11-04
Companies involved with SHG generation and products
* [http://www.advr-inc.com/waveguide.html AdvR - Frequency Conversion in KTP Waveguides]
* [http://www.radiantis.com Radiantis]