Файл: Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo .pdf

–The length of the crystal L should be compared with Le from above equations. If L < Le the respective e ect can be neglected.

4.1.2.5.2 Plane-wave fixed-field approximation

When the conditions L < Lnl and L < Le are fulfilled, the so-called fixed-field approximation is realized. For SHG, ω + ω = 2ω and ∆k = 2kω − k2ω , the conversion e ciency η is determined by the equation:

Note that all the above equations are for the SI system, i.e. [de ] = m/V ; [P ] = W ; [L] = m ; [λ] = m ; [A] = m2 ; ε0 = 8.854 × 10−12 A s/ (V m) ; c = 3 × 108 m/s .

When the powers of the mixing waves are almost equal, the conversion e ciency is for THG,

(P2 ω Pω ) 2

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In some cases (mentioned additionally) the conversion e ciency is calculated from the power (energy) of fundamental radiation, e.g. for fifth harmonic generation, ω + 4ω = 5ω :

Corresponding equations are valid for energy conversion e ciencies by substituting the pulse energy instead of power in the above equations.

The e ciency η in the case of OPO is calculated by the equation

where EOPO is the total OPO radiation energy (signal + idler) and E0 is the energy of the pump radiation. Conversion e ciency can also be determined in terms of pump depletion:

where Eunc is the energy of unconverted pumping beam after the OPO crystals. Pump depletions are usually significantly greater than the ordinary η values.

The quantum conversion e ciency (for the ratio of converted and mixing quanta) in the case of exact phase-matching (∆ k = 0) for sum-frequency generation, ω1 + ω2 = ω3 , is determined by the following equation (SI system):

In the presence of linear absorption all the above equations for conversion e ciencies should be multiplied by the factor

where α is the linear absorption coe cient of the crystal.

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4.1.2.5.3 SHG in “nonlinear regime” (fundamental wave depletion)

Analytical equation for SHG power conversion e ciency for the case of fundamental power depletion in the plane-wave approximation and for exact phase matching (∆ k = 0) is given below [99Dmi]:

4.1.3 Selection of data

Literature up to the end of 1998 is compiled in this chapter. Attempts were made to select the most reliable and recent data.

Tables in Sect. 4.1.4–4.1.8 present data on second, third, fourth, fifth, and sixth harmonic generation of Nd:YAG laser (including intracavity and in external resonant cavities), harmonic generation of iodine, ruby, Ti:sapphire, semiconductor, dye, argon, He–Ne, NH3, CO, and CO2 lasers, sum-frequency mixing (including up-conversion of IR radiation into the visible), di erencefrequency generation, optical parametric oscillation (cw, nanosecond, picosecond, and femtosecond in the UV, visible, near and mid IR regions) and picosecond continuum generation.

Second harmonic generation of Nd:YAG laser was realized with conversion e ciency of η = 80 % in KDP and KTP, THG with η = 80 % in KDP, FOHG with η = 80 − 90 % (calculated from SH) in ADP and KDP, FIHG in KDP, ADP (upon cooling) and BBO and urea (at room temperature). Second harmonic generation of Ti:sapphire laser with η = 75 % was achieved in LBO, minimum pulse durations for SH were as short as 10–16 fs (BBO, LBO). Third and fourth harmonics of Ti:sapphire laser were generated in BBO, thus covering the range of wavelengths 193–285 nm. Second harmonic of CO2 laser with η = 50 % was obtained in ZnGeP2.

Sum-frequency generation (mixing) is used, in particular, for extending the range of generating radiation into the ultraviolet. By use of SFG the shortest wavelengths in VUV were achieved with KB5 crystal (166 nm), LBO (172.7 nm), CBO, CLBO (185 nm), BBO, KDP and ADP (189, 190, and 208 nm, respectively). At present, λ = 166 nm is the minimum wavelength achieved by frequency conversion in crystals. Sum-frequency generation is also used for up-conversion of near IR (1–5 µm) and CO2 laser radiation into the visible. Maximum conversion e ciencies up to 40–60 % were obtained for the latter case in AgGaS2, CdSe, and HgGa2S4 crystals.

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Ref. p. 187] 4.1 Frequency conversion in crystals 155

Di erence-frequency generation makes it possible to produce IR radiation in the near IR (up to 7.7 µm, in LiIO3), mid IR (up to 18–23 µm, in AgGaSe2, GaSe, CdSe, Ag3AsS3) and far IR (0.05–30 mm, in LiNbO3 and GaP).

Optical parametric oscillation is a powerful method for generating continuously tunable radiation in the UV (up to 314–330 nm, in LBO and urea), visible, and IR regions (up to 16–18 µm, in CdSe and GaSe). Singly resonant OPO, or SROPO, uses resonant feedback at only the signal or idler frequency. Doubly resonant OPO, or DROPO, uses resonant feedback of both signal and idler frequencies. Exotic triply resonant OPO, or TROPO, with resonant feedback also at pump frequency, and quadruply resonant OPO, or QROPO, with SHG inside the OPO cavity and resonant feedback also at the second harmonic, are used very seldom.

Di erent OPO schemes and their energetic, temporal, spectral, and spatial characteristics are considered in detail in [73Zer, 78Dmi, 83Dan, 87Dmi] and in the three special issues of the Journal of the Optical Society of America B (Vol. 10, No. 9 and 11, 1993 and Vol. 12, No. 11, 1995) devoted to optical parametric oscillators. In Tables 4.1.30–4.1.33 we list only the main OPO parameters realized in practice: pump wavelengths, phase-matching angles, pump thresholds (peak intensity and/or average power), tuning ranges, OPO pulse durations, and conversion e ciencies for OPO experiments in the UV, visible, and near IR spectral ranges. The column headed notes gives data on the OPO type, pump intensities, crystal lengths, phase-matching temperatures, and output characteristics of OPO radiation (energy, power, bandwidth).

High conversion e ciencies were obtained with resonant schemes of cw OPO (η = 40 − 80 % with LiNbO3:MgO crystal), nanosecond (η = 60 % with BBO), traveling-wave and synchronously pumped picosecond OPO (η = 45−75 % with KDP, KTP, KTA, BBO), and synchronously pumped femtosecond OPO (η = 50 % with BBO). Minimum pulse durations were 13 fs in SP OPO with BBO crystal, pumped by the second harmonic of a Ti:sapphire laser. Very low power thresholds (0.4 mW) were achieved with LiNbO3:MgO containing quadruply resonant OPO. In general, in the case of OPO the total conversion e ciencies to both, idler and signal wavelengths, are presented. In most cases the conversion e ciency corresponds to the maximum for the range of wavelengths.

The picosecond continuum, first detected in media with cubic nonlinearity (D2O, H2O, etc.), was also observed in crystals with square nonlinearity (KDP, LiIO3, LiNbO3, etc.).

We don’t pretend to comprehend all directions of frequency conversion in crystals. Some special aspects, e.g. second harmonic generation in layers and films, waveguides and fibers, periodically poled crystals, liquid crystals, as well as di erent design configurations of frequency converters have been beyond our consideration. For “justification” we refer to Artur L. Schawlow’s famous saying: “To do successful research, you don’t need to know everything. You just need to know of one thing that isn’t known”.

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