Can somebody explain meaning of this term in simple terms and give equation describing it? Thank you in advance!
The number of concepts in physics is larger than number of words....So, coherence length in linear optics and coherence length in nonlinear optics are different concepts. NLO coherence length is related to the case when fundamental and generated waves propagate along nonlinear medium, and phase difference between at the entry is, say, zero, and after passing the coherence length it becomes equal to pi. Then, according to wave equation solution, generated waves become "fundamental" in some sense, and vice versa. I.e., generated waves begin to be converted back to fundamental ones when phase difference is between pi and 2pi.
The maximum value of the amplitude of the second (third) harmonic is achieved on the length of the nonlinear crystal lcoh = π /Δk, Δk - is the wavevector mismatch between the fundamental and second (third) harmonic waves in a nonlinear material.
At the same length there is a reduction of the harmonic amplitude maximum value to zero. Length of lcoh is called the coherence length.
http://en.wikipedia.org/wiki/Second-harmonic_generation
http://users.uj.edu.pl/~ufdzierz/PracFot/SHG_Dood.pdf
http://evgenynn.narod.ru/publications/19_Extremely.pdf
Best regards, Leonid
The coherence length is the maximum length Lcoh over which the second/third harmonic generated at some point along the medium still adds constructively with the second/third harmonic previously generated at a distance L earlier in the medium.
To be more precise: the phase of the second/third harmonic generated at any point is always in determined by the phase of the fundamental. However, dispersion in the medium causes the generated second/third harmonic to propagate with a different phase velocity with the fundamental. Therefore after some propagation distance L, the second/third harmonic generated will add destructively with the second/third harmonic previously generated, causing back-conversion (i.e. the second/third harmonic will be converted back to the fundamental). This occurs at the coherence length Lcoh = π /Δk as previously mentioned. As a result, the intensity of the second/third harmonic oscillates along the medium with twice this periodicity (such that after 2Lcoh, all the harmonic energy is transferred back to the medium).
Phase matching is used to ensure that the phase velocities of the harmonic and fundamental fields are the same. If this is not possible, quasi phase matching can be used. This can be achieved (all nonlinear processes) by turning off the harmonic generation during the periods when the fields are out of phase (used in high harmonic generation). For even-order harmonic generation (e.g. second), the sign of the polarization can be reversed such that the sign of the generated harmonic is inverted, thus destructive interference becomes constructive!
In classical view, phase matching in nonlinear optics is required because of dispersion. The higher harmonic is generated by the fundamental field that propagates with a velocity of c/n(/omega). The nonlinear radiation however propagates with a velocity of c/n(2/j * omega) where j is an integer which can be 2 (SHG), 3 (THG) so forth. Therefore the velocity of the phase that generates nonlinear dipole radiation and the velocity of the nonlinear radiation is generally not the same unless it is phase matched. For certain lengths known as the coherene length the two phases still add up constructively. Afterwards they interfere destructively, then coherently again so that the the nonlinear intensity profile is sinusoidal for the case of non-phase matched. The length of coherence depends on the difference of the two phase velocity or in other words depends on how much the refractive index between n(/omega) and n(j omega) differs. For linear optics the dipole radiation and fundamental source propagates at the same velocity because they both feel the same refractive index n(/omega) so phase matching inside the material is always achieved.
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