{"id":10866,"date":"2022-10-03T02:06:37","date_gmt":"2022-10-03T02:06:37","guid":{"rendered":"https:\/\/www.szlaser.com\/?page_id=10866"},"modified":"2022-10-04T02:19:26","modified_gmt":"2022-10-04T02:19:26","slug":"diode-pumped-ndyag-green-laser","status":"publish","type":"page","link":"https:\/\/www.szlaser.com\/index.php\/wiki\/diode-pumped-ndyag-green-laser\/","title":{"rendered":"Diode-Pumped Nd:YAG Green Laser with Q-Switch and Mode Locking"},"content":{"rendered":"<div class=\"fusion-fullwidth fullwidth-box fusion-builder-row-1 fusion-flex-container has-pattern-background has-mask-background nonhundred-percent-fullwidth non-hundred-percent-height-scrolling\" style=\"--awb-border-radius-top-left:0px;--awb-border-radius-top-right:0px;--awb-border-radius-bottom-right:0px;--awb-border-radius-bottom-left:0px;--awb-flex-wrap:wrap;\" ><div class=\"fusion-builder-row fusion-row fusion-flex-align-items-flex-start fusion-flex-content-wrap\" style=\"max-width:1331.2px;margin-left: calc(-4% \/ 2 );margin-right: calc(-4% \/ 2 );\"><div class=\"fusion-layout-column fusion_builder_column fusion-builder-column-0 fusion_builder_column_1_1 1_1 fusion-flex-column\" style=\"--awb-bg-size:cover;--awb-width-large:100%;--awb-margin-top-large:0px;--awb-spacing-right-large:1.92%;--awb-margin-bottom-large:20px;--awb-spacing-left-large:1.92%;--awb-width-medium:100%;--awb-order-medium:0;--awb-spacing-right-medium:1.92%;--awb-spacing-left-medium:1.92%;--awb-width-small:100%;--awb-order-small:0;--awb-spacing-right-small:1.92%;--awb-spacing-left-small:1.92%;\"><div class=\"fusion-column-wrapper fusion-column-has-shadow fusion-flex-justify-content-flex-start fusion-content-layout-column\"><div class=\"fusion-title title fusion-title-1 fusion-sep-none fusion-title-text fusion-title-size-one\"><h1 class=\"fusion-title-heading title-heading-left fusion-responsive-typography-calculated\" style=\"margin:0;--fontSize:32;line-height:1.3;\"><h1>Diode-Pumped Nd:YAG Green Laser with Q-Switch and Mode Locking<\/h1><\/h1><\/div><div class=\"fusion-text fusion-text-1\"><p>Gaining of a high peak power in the visible light range from a solid state laser with continuous diode pumping is a challenging task in several applications (high-precision material processing, nonlinear optics and Raman-spectroscopy, medicine, etc.). The technique of modulating the Q-factor of laser cavity (Q-switch) enables growth of the peak power approximately as \u03c4sp \/ \u03c4ph (here \u03c4sp is the upper laser level lifetime and \u03c4ph is the photon lifetime in the cavity).<\/p>\n<p>When we take a typical Nd:YAG laser, this gain in peak power is about 103- 104 times. The further growth of the peak power is possible through methods of mode locking of the laser (ML).<\/p>\n<p>However, realization of mode-locking together with Q-switch (unlike the case of continuous operation mode) is a technically challenging task: we face a high amplification, almost uncontrollable nonlinear effects, damage of optical elements in the laser, etc.<\/p>\n<p>In prior art, the steady mode of generation for Q-switch coupled with mode locking (so called QML) is accomplished with using of two acousto-optic modulators (AOM) in a cavity; one modulator operates in the traveling acoustic wave, and the other modulator has the standing wave (see, e.g., Kuizenga, 1981).<\/p>\n<p>Actually, the QML mode is possible when we introduce absorbing elements within the cavity (Herrmann &amp; Wilhelmi, 1987; He et al., 1996; Chen et al., 2001; Agnesi et al., 2001; Pan et al., 2007), but in this case the higher pumping level means a higher pulse repetition rate.<\/p>\n<p>Therefore the practical levels of peak power are very moderate. In this chapter we inform about a new technique for obtaining a steady QLM lasing mode in a diode-pumped green Nd:YAG laser; this type of operation is possible with only one travelling-wave AOM. As for the further shortening of lasing pulse duration (\u0394\u03c4), it is achieved through formation of a Kerr lens in a doubling-frequency crystal. Since the process of make-up a Kerr lens is very rapid, this trick gives us \u0394\u03c4 as small as the theoretical limit \u2248 1\/\u0394\u03bd (here \u0394\u03bd is the spectral bandwidth of generation).<\/p>\n<\/div><div class=\"fusion-title title fusion-title-2 fusion-sep-none fusion-title-text fusion-title-size-three\"><h3 class=\"fusion-title-heading title-heading-left fusion-responsive-typography-calculated\" style=\"margin:0;--fontSize:24;line-height:1.3;\"><h3><span class=\"notion-enable-hover\" data-token-index=\"0\" data-reactroot=\"\">The general scheme of experiment and key parameters of Nd:YAG laser<\/span><\/h3><\/h3><\/div><div class=\"fusion-text fusion-text-2\"><p>Now the class of solid-state lasers demonstrate a rebirth due to replacement of pumping with arc lamps by diode-laser pumping; this ensures a higher efficiency of the device (higher than 10%) and a longer service life (more than 10,000 hrs).<\/p>\n<p>Typically, diode laser (DL) can be used in different schemes of pumping \u2013 longitudinal or transversal. Solid-state lasers with longitudinal pumping are more efficient and the emission beam has better quality (\u041c2 \u2248 1) (Tidwell et al.,1992; Tsunekane et al., 1998).<\/p>\n<p>However, it is very difficult to gain a high output power at this geometry of pumping. When the required output power is above 100 W, the good choice is lasers with transverse pumping of the active element (e.g., in Ostermeyer et al., 2002; Fujikawa et al., 2001; Lee et al., 2002). But the efficiency of this scheme is lower and lasing with \u041c2 \u2248 1 usually faces serious difficulties.<\/p>\n<p>In this paragraph we describe the experimental scheme of a Nd:YAG laser with transverse pumping by DL arrays and with time-averaged output power of 15 W (in \u0422\u0415\u041c00-mode).<\/p>\n<p>When the effective doubling of laser emission frequency is achieved, this scheme enables lasing with the output power of 12 W (here the wavelength is \u03bb = 532 nm). The laser scheme with a Z-shaped and 4-mirrored cavity (Donin et al., 2004) is depicted in Fig.1. The reflection coefficient (r) for mirrors \u041c1-\u041c4 at the wavelength 1064 nm was better than 99.5%. Mirror \u041c4 was a dichroic mirror with r &gt; 99.5 % (at \u03bb=532 nm); at the latter wavelength the transparency coefficient (T) for mirror \u041c3 was 92 %.<\/p>\n<p>To gain the light generation at the second harmonics between spherical mirrors M3 and M4 (they have curvature radii R =200 and 150 mm, correspondingly), a nonlinear crystal (\u0412\u0412\u041e, \u041a\u0422\u0420 or LBO) was placed into the zone of cavity waist . The beam diameter in this zone was about 100 -150 \u00b5m. To gain a maximum power of emission at the wavelength \u03bb=1064 nm, the mirror \u041c1 was replaced with a mirror possessing an optimal transmission coefficient (but there was no need in the nonlinear crystal within the resonator).<\/p>\n<p>The active element was a Nd<sup>3+<\/sup> :YAG crystal (mass concentration for Nd was 1 %) with the diameter of 2 mm and the length of 63 mm; this element was illuminated from three sides with bars of LD arrays emitting at the wavelength \u03bb \u0440 =808 nm (Fig. 1b). The laser\u2019s active element together with DLs is cooled by distilled water flow. This cooling is accomplished by a circulation-type close-loop cooler; this device ensures temperature stability within 0.1 0\u0421.<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-fusion-600 wp-image-10868\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-1.-Scheme-a-and-pump-geometry-b-600x326.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-1.-Scheme-a-and-pump-geometry-b-600x326.png\" alt=\"\" width=\"600\" height=\"326\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27600%27%20height%3D%27326%27%20viewBox%3D%270%200%20600%20326%27%3E%3Crect%20width%3D%27600%27%20height%3D%27326%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-1.-Scheme-a-and-pump-geometry-b-200x109.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-1.-Scheme-a-and-pump-geometry-b-300x163.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-1.-Scheme-a-and-pump-geometry-b-400x217.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-1.-Scheme-a-and-pump-geometry-b-600x326.png 600w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-1.-Scheme-a-and-pump-geometry-b-768x417.png 768w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-1.-Scheme-a-and-pump-geometry-b-800x435.png 800w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-1.-Scheme-a-and-pump-geometry-b.png 944w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 1. Scheme (a) and pump geometry (b) of an Nd:YAG laser: M1 \u2013 M4 \u2013 resonator mirrors, qw \u2013quarzrotator.<\/h5>\n<\/div><div class=\"fusion-text fusion-text-3\"><p>The operation temperature of laser was close to 28 0\u0421 and it can be regulated in a wide range. The dependency of laser output power on the coolant water temperature is plotted in Fig.2. The Q-factor of laser was modulated with the MZ-305 type AOM with the travelling acoustic wave. The AOM with the driving frequency of 50 MHz was manufactured from crystalline quartz and equipped with water cooling facility. The quarzrotator 5 could be used for compensation of induced birefringence.<\/p>\n<p>The laser resonator was calculated with the matrix method with account for a heat lens in the<\/p>\n<p>active element and dispersion of the air gap between the nonlinear crystal and mirror 4. We have also estimated the diffraction losses of \u0422\u0415\u041c<sub>\u043e\u043e<\/sub>-mode (\u2248 10%) and higher-order modes, e.g., for T\u0415M<sub>01<\/sub> mode (\u2248 50%). All these losses ensured lasing in the regime close to \u0422\u0415\u041c<sub>\u043e\u043e<\/sub>-mode.<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10869\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Output-laser-power-vs.-temperature.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Output-laser-power-vs.-temperature.png\" alt=\"\" width=\"522\" height=\"369\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27522%27%20height%3D%27369%27%20viewBox%3D%270%200%20522%20369%27%3E%3Crect%20width%3D%27522%27%20height%3D%27369%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Output-laser-power-vs.-temperature-200x141.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Output-laser-power-vs.-temperature-300x212.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Output-laser-power-vs.-temperature-400x283.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Output-laser-power-vs.-temperature.png 522w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 522px) 100vw, 522px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 2. Output laser power vs. temperature.<\/h5>\n<\/div><div class=\"fusion-text fusion-text-4\"><p>When the dimensions of the active element were chosen in the view of most effective usage of this element, we took into consideration the following things.\u00a0During of active element pumping, the birefringence effect takes place: this causes depolarization of beam passing the crystal and makes the losses higher. The depolarization-caused losses are proportional to \u03c94, and the output power has the law \u03c92 ( here \u03c9 is the radius of \u0422\u0415\u041c00-mode in active element).<\/p>\n<p>This means that we can find the optimal value for the resonator&#8217;s fundemental mode. The estimate for optimal \u03c9 for our conditions was about \u2248 1 mm; this coincides with results from (Matthew et al., 1996). Therefore the optimal diameter of rod should be take \u2248 2 mm.<\/p>\n<p>The dependencies of output power (Pout) of the laser at \u03bb=1064 nm (at the AOM switching frequency or the pulse repetition rate F = 20 kHz) on the current (I) through DL, as well as function \u043ef emitted light power (Plight) and consumed electric power (\u0420electr) of DLs are plotted in Fig.3. The saturation of Pout along with current growth is caused by the resonator mismatch due to thermo-optic effects.<\/p>\n<p>The thermal lens in active element has the focal distance (at saturation current) close to 25 cm. The efficiency coefficient on the total electric power was 5%, and the light efficiency was 12.5 % (the differential coefficients of efficiency were 10% and 21%, correspondingly). These curves were obtained for a configuration with an exit mirror M1 (see Fig.1) which has the optimal transmission coefficient \u0422 = 20 %. Under described conditions, generation took place in the \u0422\u0415\u041c\u043e\u043e-mode.<\/p>\n<p>Fig.4 shows the plotting of the output power and the lasing pulse duration \u0394\u03c4 vs. the pulse repetition rate (F). The pulse length was measured with an avalanche photodiode type LFD-2 with the time resolution better than 1 ns.<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-fusion-600 wp-image-10870\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-3.-Dependence-of-the-1064-nm-output-power-on-the-current-600x454.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-3.-Dependence-of-the-1064-nm-output-power-on-the-current-600x454.png\" alt=\"\" width=\"600\" height=\"454\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27600%27%20height%3D%27454%27%20viewBox%3D%270%200%20600%20454%27%3E%3Crect%20width%3D%27600%27%20height%3D%27454%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-3.-Dependence-of-the-1064-nm-output-power-on-the-current-200x151.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-3.-Dependence-of-the-1064-nm-output-power-on-the-current-300x227.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-3.-Dependence-of-the-1064-nm-output-power-on-the-current-400x303.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-3.-Dependence-of-the-1064-nm-output-power-on-the-current-600x454.png 600w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-3.-Dependence-of-the-1064-nm-output-power-on-the-current.png 676w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 3. Dependence of the 1064 nm output power on the current, light output power and the consumed electric power of the DL.<\/h5>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-fusion-600 wp-image-10871\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-4.-Dependence-of-the-output-power-at-1064-nm-600x473.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-4.-Dependence-of-the-output-power-at-1064-nm-600x473.png\" alt=\"\" width=\"600\" height=\"473\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27600%27%20height%3D%27473%27%20viewBox%3D%270%200%20600%20473%27%3E%3Crect%20width%3D%27600%27%20height%3D%27473%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-4.-Dependence-of-the-output-power-at-1064-nm-200x158.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-4.-Dependence-of-the-output-power-at-1064-nm-300x236.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-4.-Dependence-of-the-output-power-at-1064-nm-400x315.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-4.-Dependence-of-the-output-power-at-1064-nm-600x473.png 600w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-4.-Dependence-of-the-output-power-at-1064-nm-768x605.png 768w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-4.-Dependence-of-the-output-power-at-1064-nm-800x631.png 800w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-4.-Dependence-of-the-output-power-at-1064-nm.png 940w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 4. Dependence of the output power at 1064 nm and the laser pulse duration on the pulse repetition rate.<\/h5>\n<\/div><div class=\"fusion-title title fusion-title-3 fusion-sep-none fusion-title-text fusion-title-size-three\"><h3 class=\"fusion-title-heading title-heading-left fusion-responsive-typography-calculated\" style=\"margin:0;--fontSize:24;line-height:1.3;\"><h3>Frequency doubling for laser emission<\/h3><\/h3><\/div><div class=\"fusion-text fusion-text-5\"><p>Let us focus on the problem of frequency doubling (second harmonic generation) for laser emission. It is known (Smith, 1970) that while intra-cavity frequency doubling the losses brought by a nonlinear crystal must be balanced to the losses brought by an optimal mirror for a laser without nonlinear crystal. This condition enables a maximum output power at the doubled frequency.<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10872\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-5.-Laser-output-power-.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-5.-Laser-output-power-.png\" alt=\"\" width=\"435\" height=\"422\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27435%27%20height%3D%27422%27%20viewBox%3D%270%200%20435%20422%27%3E%3Crect%20width%3D%27435%27%20height%3D%27422%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-5.-Laser-output-power--200x194.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-5.-Laser-output-power--300x291.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-5.-Laser-output-power--400x388.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-5.-Laser-output-power-.png 435w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 435px) 100vw, 435px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 5. Laser output power (\u03bb=1064 nm) vs. transmission of output mirror T.<\/h5>\n<p>Fig.5 presents an experimental plotting of laser output power as a function of transmission coefficient for the output Mirror (DL current I = 17A).<\/p>\n<p>In choosing the type of nonlinear crystals for the second harmonic generation and picking up optimal parameters (including the crystal length dcr ), we made computer simulations for the double-frequency output power. These simulations started from a system of truncated equations for the real parts of electric field amplitudes; this system of equations was derived from the wave equation by the method of slowly varying amplitude in low-absorption and low-nonlinearity medium within the plain wave approximation.<\/p>\n<p>These equations were solved by using software specially developed for determining the all required parameters of nonlinear crystals (even in the case of deviation from precise phase matching). We applied a simple differential scheme for a particular case of second harmonics generation when effects with normalized phase (\u0394kz + \u03c0\/2) accumulates most rapidly. At the initial conditions z=0, a2(0)=0 we solved numerically this equation system (Dmitriev &amp; Tarasov, 1982)<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10873\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-\u03c3a1-\u03c3a2.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-\u03c3a1-\u03c3a2.png\" alt=\"\" width=\"266\" height=\"146\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27266%27%20height%3D%27146%27%20viewBox%3D%270%200%20266%20146%27%3E%3Crect%20width%3D%27266%27%20height%3D%27146%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-\u03c3a1-\u03c3a2-200x110.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-\u03c3a1-\u03c3a2.png 266w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 266px) 100vw, 266px\" \/><\/p>\n<p>For the case of precise phase matching (wave detuning \u0394k*=*2k1-k2=0) this equation system\u00a0has an exact analytical solution:<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10874\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-a2.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-a2.png\" alt=\"\" width=\"428\" height=\"48\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27428%27%20height%3D%2748%27%20viewBox%3D%270%200%20428%2048%27%3E%3Crect%20width%3D%27428%27%20height%3D%2748%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-a2-200x22.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-a2-300x34.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-a2-400x45.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-a2.png 428w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 428px) 100vw, 428px\" \/><\/p>\n<p>Here \u03c31=4\u03c02deff\/\u03bb1n1, \u03c32=2\u03c02deff\/\u03bb2n2 are the nonlinear coupling coefficients, deff is the\u00a0effective nonlinearity, \u03bbi is the wavelength, ni is the refraction index, ki is the wave vector, \u0430i\u00a0is the electric field amplitude, \u043010 is the initial amplitude of the electric field for the\u00a0fundemental emission, z is the running coordinate.<\/p>\n<\/div><div class=\"fusion-text fusion-text-6\"><p>In these calculations we took the formulae and input data for calculating the effective nonlinearity deff and Sellmeyer\u2019s equations for determining the refraction index ni as in (Linet al., 1990; Gurzadyan et al., 1991). The phase-matching angles for the wavelengths of fundemental radiation (1.07-0.75 \u00b5m) were calculated by well-known formulae (see\u00a0Dmitriev &amp; Tarasov, 1982).<\/p>\n<p>These calculations were performed for nonlinear crystals LBO, \u041a\u0422\u0420, \u0412\u0412\u041e. Fig. 6 presents calculated the second-harmonic power density S2 as a function of the crystal length. We took the power density of fundamental input radiation S1(0) equal to 150 MW\/cm2, which corresponds to the Q-switch frequency F = 10-15 kHz. The power density was defined as Si = c ni \u0430i2\/8\u03c0, here c is the velocity of light in vacuum. The approximate lengths of crystals which are optimal for generating output power at the second harmonics are the following: \u2248 10 mm (LBO, phase-matched type I), \u2248 2 mm (\u041a\u0422\u0420, phase-matched type II), and \u2248 5 mm (\u0412\u0412\u041e, phase-matched type I ).<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-fusion-600 wp-image-10875\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-6.-Calculated-dependences-of-the-second-harmonic-power-densities-S2-600x340.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-6.-Calculated-dependences-of-the-second-harmonic-power-densities-S2-600x340.png\" alt=\"\" width=\"600\" height=\"340\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27600%27%20height%3D%27340%27%20viewBox%3D%270%200%20600%20340%27%3E%3Crect%20width%3D%27600%27%20height%3D%27340%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-6.-Calculated-dependences-of-the-second-harmonic-power-densities-S2-200x113.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-6.-Calculated-dependences-of-the-second-harmonic-power-densities-S2-300x170.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-6.-Calculated-dependences-of-the-second-harmonic-power-densities-S2-400x227.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-6.-Calculated-dependences-of-the-second-harmonic-power-densities-S2-600x340.png 600w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-6.-Calculated-dependences-of-the-second-harmonic-power-densities-S2-768x435.png 768w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-6.-Calculated-dependences-of-the-second-harmonic-power-densities-S2-800x453.png 800w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-6.-Calculated-dependences-of-the-second-harmonic-power-densities-S2.png 944w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 6. Calculated dependences of the second-harmonic power densities S2 on the crystal length.<\/h5>\n<p>The walk-off angle was calculated by formula<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10876\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/formula-tan.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/formula-tan.png\" alt=\"\" width=\"398\" height=\"97\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27398%27%20height%3D%2797%27%20viewBox%3D%270%200%20398%2097%27%3E%3Crect%20width%3D%27398%27%20height%3D%2797%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/formula-tan-200x49.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/formula-tan-300x73.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/formula-tan.png 398w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 398px) 100vw, 398px\" \/><\/p>\n<p>where \u03b8 is the phase-matching angle, no and ne \u2013 ordinary and extraordinary refractive index. The same software was used for estimating of the angular and spectral phasematching width.<\/p>\n<\/div><div class=\"fusion-text fusion-text-7\"><p>In our experiment on study of the second harmonics generation, we used the \u0412\u0412\u041e crystals (phase-matched type I: \u03b8=22.80, \u03c6=900; dcr =5 mm), \u041aTP (phase-matched type II: \u03b8=900, \u03c6=23.50; dcr =5 mm), and LBO crystal (phase-matched type I: \u03b8=900, \u03c6=11.60, dcr =12 mm).<\/p>\n<p>All the tested crystals had two-hump antireflection coatings at the operating wavelength with r &lt; 0.5%. This allowed us to obtain the maximal power of generation: \u0412\u0412\u041e \u2013 5 W (F=15 kHz), \u041a\u0422\u0420 \u2013 12 W (F &gt;20 kHz), and LBO \u2013 8.3 W (F = 10 kHz). We should note that for the BBO crystal the maximal power was restricted by coating destruction.<\/p>\n<p>A typical dependence of second-harmonic output power on the DL current for a LBO crystal is shown in Fig.7. One can see that at the initial interval, this dependency has the second power law, but later it goes slower due to resonator mismatch caused by thermo-optic effects. The curve of output power vs. the pulse repetition rate frequency for an LBO crystal had a maximum at the F = 10 kHz.<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-fusion-600 wp-image-10877\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-7.-Dependence-of-the-second-harmonic-output-power--600x427.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-7.-Dependence-of-the-second-harmonic-output-power--600x427.png\" alt=\"\" width=\"600\" height=\"427\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27600%27%20height%3D%27427%27%20viewBox%3D%270%200%20600%20427%27%3E%3Crect%20width%3D%27600%27%20height%3D%27427%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-7.-Dependence-of-the-second-harmonic-output-power--200x142.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-7.-Dependence-of-the-second-harmonic-output-power--300x214.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-7.-Dependence-of-the-second-harmonic-output-power--400x285.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-7.-Dependence-of-the-second-harmonic-output-power--600x427.png 600w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-7.-Dependence-of-the-second-harmonic-output-power--768x546.png 768w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-7.-Dependence-of-the-second-harmonic-output-power--800x569.png 800w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-7.-Dependence-of-the-second-harmonic-output-power-.png 1015w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 7. Dependence of the second-harmonic output power P2\u03bd on the DL current I for an LBO crystal at F = 10 kHz.<\/h5>\n<p>For a KTP crystal, the generated power is growing steadily up to F=20 kHz, and becomes constant in the interval from 20 to 40 kHz. For a KTP crystal, the coefficient of conversion into second harmonic was 80 %. For LBO the maximal power of second harmonic was obtained at driving frequency of 10 kHz, and this corresponds to 10.5 W generation at wavelength \u03bb=1064 nm (Fig.7). Therefore, the coefficient of conversion into second harmonic for this type of crystal was 79%.<\/p>\n<p>Fig.8 presents the dependency of pulse energy and peak power at the wavelength 532 nm as a function of Q-switch frequency (the case of LBO crystal and a cavity on Fig. 9).<\/p>\n<p>Our results demonstrated the efficient conversion of the fundamental frequency into the second harmonic for the case of Q-switched pulse energy \u2248 1 mJ; this gives hope for developing an effective laser with output at wavelength \u03bb = 532 nm and with the average output power for \u0422\u0415\u041c00-mode as high as 100 W and even higher.<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10878\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-8.-Dependence-of-pulse-energy-Ep-1.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-8.-Dependence-of-pulse-energy-Ep-1.png\" alt=\"\" width=\"503\" height=\"400\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27503%27%20height%3D%27400%27%20viewBox%3D%270%200%20503%20400%27%3E%3Crect%20width%3D%27503%27%20height%3D%27400%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-8.-Dependence-of-pulse-energy-Ep-1-177x142.png 177w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-8.-Dependence-of-pulse-energy-Ep-1-200x159.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-8.-Dependence-of-pulse-energy-Ep-1-300x239.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-8.-Dependence-of-pulse-energy-Ep-1-400x318.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-8.-Dependence-of-pulse-energy-Ep-1.png 503w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 503px) 100vw, 503px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 8. Dependence of pulse energy Ep (1) and power Pout (2) at \u03bb = 532 nm on the pulse repetition rate F.<\/h5>\n<\/div><div class=\"fusion-title title fusion-title-4 fusion-sep-none fusion-title-text fusion-title-size-three\"><h3 class=\"fusion-title-heading title-heading-left fusion-responsive-typography-calculated\" style=\"margin:0;--fontSize:24;line-height:1.3;\"><h3>Q-switch and mode locking achieved with a travelling-wave AOM<\/h3><\/h3><\/div><div class=\"fusion-text fusion-text-8\"><p>Here we develop a new method (Donin et al., 2011) of QLM lasing; the design includes a end-spherical mirror (SM) in the cavity and one travelling-wave AOM (abbreviation \u2013 SMAOM). The operation principle for SMAOM configuration and laser scheme is illustrated in Fig. 9. The Nd:YAG green laser was assembled by the key diagram shown in Fig.1.<\/p>\n<p>The mirror curvature radii were 200, -900, 200 and 150 mm accordingly for the mirrors M1, M2, M3 and M4. The cavity optic length was L =1.5 m. The \u0410\u041e\u041c was placed at the Bragg angle (\u03b8B) to the optic axis of cavity near the terminal spherical mirror \u041c1.<\/p>\n<p>The AOM\u2019s center is distanced from the mirror\u2019s reflecting surface by distance R1, which is the mirror curvature radius. When the driving frequency is f = 50 MHz, which equals a half of laser\u2019s intermode interval c\/2L =2f , a travelling acoustic wave is created in the quartz block of AOM (it is shown by small bold-symbol arrow in Fig. 9); this travelling wave is a source for Bragg\u2019s diffraction for laser emission. Since the light beam (with frequency \u03bd0) passes through the AOM from right to left, two beams enter the mirror (1 and 2).<\/p>\n<p>Beam 1 goes along the cavity axis and being reflected from the mirror backward without any change in initial frequency \u03bd0. The beam 2 feels the Bragg diffraction and travels to the same mirror with frequency (\u03bd0+f). The reflects from the spherical surface of the mirror and returns to the AOM, where it is split into a beam with the unchanged frequency (\u03bd0+ f), exiting the cavity in the backward direction at the angle 2\u03b8B, and another beam after repeated diffraction (on the AOM\u2019s quartz block). The latter kind of beam has a frequency (\u03bd0 + 2f) and travels backward along the cavity axis. This beam produces the effect of mode locking.<\/p>\n<p>The beam with frequency (\u03bd0 + f) exiting the cavity at angle 2\u03b8B provides modulating of losses in the cavity; so the laser operates in the mode of Q-switch with the pulse repetition rate which is given by the AOM switching frequency (F~1\u00f7100 kHz).<\/p>\n<p>The good feature of this scheme is that when the driving frequency is off, the acoustic wave inside then AOM dies during a time t = db\/Vsnd = 0.2cm\/5\u00b7105 cm\/s \u2248 0.4 \u00b5s (here db is the laser beam diameter in the quartz block, Vsnd is the sound velocity in the material). The duration of the lasing pulse in the Q-switch mode was \u2248 100 ns, i.e., during time t due to the beam of repeated diffraction with frequency (\u03bd0+ 2f) mode locking is achieved in the generation pulse.<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-fusion-600 wp-image-10879\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-9.-Laser-diagram-and-operation-principle-for-SMAOM-600x288.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-9.-Laser-diagram-and-operation-principle-for-SMAOM-600x288.png\" alt=\"\" width=\"600\" height=\"288\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27600%27%20height%3D%27288%27%20viewBox%3D%270%200%20600%20288%27%3E%3Crect%20width%3D%27600%27%20height%3D%27288%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-9.-Laser-diagram-and-operation-principle-for-SMAOM-200x96.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-9.-Laser-diagram-and-operation-principle-for-SMAOM-300x144.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-9.-Laser-diagram-and-operation-principle-for-SMAOM-400x192.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-9.-Laser-diagram-and-operation-principle-for-SMAOM-600x288.png 600w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-9.-Laser-diagram-and-operation-principle-for-SMAOM-768x369.png 768w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-9.-Laser-diagram-and-operation-principle-for-SMAOM-800x385.png 800w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-9.-Laser-diagram-and-operation-principle-for-SMAOM.png 982w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 9. Laser diagram and operation principle for SMAOM.<br \/>\n\u041c1-\u041c4 \u2013 cavity mirrors, \u0410\u041e\u041c &#8211; acoustic-optic modulator, Nd: YAG \u2013 active element, LBO \u2013 nonlinear crystal, D \u2013 diaphragm<\/h5>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10880\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-10.-Oscillogram-of-generated-pulse-at-the-wavelength-\u03bb-1064-nm.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-10.-Oscillogram-of-generated-pulse-at-the-wavelength-\u03bb-1064-nm.png\" alt=\"\" width=\"463\" height=\"268\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27463%27%20height%3D%27268%27%20viewBox%3D%270%200%20463%20268%27%3E%3Crect%20width%3D%27463%27%20height%3D%27268%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-10.-Oscillogram-of-generated-pulse-at-the-wavelength-\u03bb-1064-nm-200x116.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-10.-Oscillogram-of-generated-pulse-at-the-wavelength-\u03bb-1064-nm-300x174.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-10.-Oscillogram-of-generated-pulse-at-the-wavelength-\u03bb-1064-nm-400x232.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-10.-Oscillogram-of-generated-pulse-at-the-wavelength-\u03bb-1064-nm.png 463w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 463px) 100vw, 463px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 10. Oscillogram of generated pulse at the wavelength \u03bb = 1064 nm produced in QML mode.<br \/>\nThe division value for abscissa axis is 50 ns.<\/h5>\n<p>On the first stage, we performed measurements without nonlinear crystal and diaphragm (this means no frequency doubling and no Kerr lens formation). In this case we replaced the mirror \u041c1 with another one with similar curvature radius, but possessing the transparency\u00a0of \u0422 = 11% at \u03bb = 1064 nm.<\/p>\n<p>The oscillogram of a Q-switch pulse with mode locking is depicted in Fig.10. The average power of laser was 2 W (at the Q-switch frequency equal to 2 kHz). The registration system resolution time (photodiode and oscillograph) \u2248 2 ns did not allow us to determine the actual duration of pulses inside a &#8220;train&#8221;, so this task had required assembling an optical correlator for pulse registration on the second harmonic in the KTP crystal (collinear scheme). This optical correlator gave the pulse duration for mode\u00a0locking generation about 40 ps (see Fig. 14 \u0430), i.e., a single peak power was \u2248 2 MW.<\/p>\n<\/div><div class=\"fusion-title title fusion-title-5 fusion-sep-none fusion-title-text fusion-title-size-three\"><h3 class=\"fusion-title-heading title-heading-left fusion-responsive-typography-calculated\" style=\"margin:0;--fontSize:24;line-height:1.3;\"><h3>The Kerr lens based on nonlinear crystal<\/h3><\/h3><\/div><div class=\"fusion-text fusion-text-9\"><p>The further gain in reducing a single pulse duration and growth of its peak power has been achieved using a Kerr lens and a diaphragm (laser generation at the line \u03bb = 532 nm as shown in Fig. 9). Kerr lens is formed in a nonlinear crystal for 2nd harmonic generation\u00a0(LBO, phase-matched type I, the length of dcr = 20 mm). For the first time, the phenomena of mode locking with a Kerr lens (or self-locking of mode) has been studied in (Spence et al., 1991). The foundation of this method is a radiation self-focusing in a media; this opens the\u00a0way to creating an affect similar to absorbing elements in a laser cavity.<\/p>\n<p>Self-focusing develops due to dependency of material refraction index on radiation intensity: n = n0 + n2I.<br \/>\nThis effect creates a lens with focus depending on light intensity. This type of nonlinear lens matched with a diaphragm works as a kind of saturable absorber. Another design is too possible: the same lens without a diaphragm, and the cavity elements work as a diaphragm.<br \/>\nIn the case when self-focusing is driven by electron polarization under impact of the field of light wave, this would create an almost zero-inertia \u201csaturable absorber\u201d &#8211; the response time is ~10-15s (see, e.g., Shapiro, 1977). The calculating of cavity\u2019s parameters was done with the matrix method. The following matrix was used for describing the beam passing through the Kerr element (Magni et al., 1993):<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10881\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-M.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-M.png\" alt=\"\" width=\"325\" height=\"69\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27325%27%20height%3D%2769%27%20viewBox%3D%270%200%20325%2069%27%3E%3Crect%20width%3D%27325%27%20height%3D%2769%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-M-200x42.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-M-300x64.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-M-320x69.png 320w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-M.png 325w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 325px) 100vw, 325px\" \/><\/p>\n<p>where de = dcr\/n0 is the effective length of media at the inter-resonator power \u0420 = 0, and here<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10882\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-rp.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-rp.png\" alt=\"\" width=\"317\" height=\"86\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27317%27%20height%3D%2786%27%20viewBox%3D%270%200%20317%2086%27%3E%3Crect%20width%3D%27317%27%20height%3D%2786%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-rp-200x54.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-rp-300x81.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-rp.png 317w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 317px) 100vw, 317px\" \/><\/p>\n<p>where p = \u0420\/\u0420\u0441 (Pc = c\u03b50\u03bb2\/(2\u03c0n2) is the critical power for self-focusing), \u03c9\u0441 is the beam size in the middle of medium, and \u03c90 is the beam size in the waist, calculated at p = 0. The effect of \u201csaturable absorber\u201d requires a shrinking of the beam size with its intensity growing in the plane of the diaphragm. Quantitatively, this effect is described by a parameter (Magni et al., 1993)<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10883\" src=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27158%27%20height%3D%2788%27%20viewBox%3D%270%200%20158%2088%27%3E%3Crect%20width%3D%27158%27%20height%3D%2788%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/equation-of-\u03b4.png\" alt=\"\" width=\"158\" height=\"88\" \/><\/p>\n<p>where \u03c9 is the radius of Gaussian beam in the specific plane within the cavity. To obtain the shorter pulse, parameters \u03b4 should be negative and high in modulus. The diaphragm was placed in the plane at the end-mirror \u041c4 (see Fig. 9). The variable parameter in our simulations was the distance between the end-mirror \u041c4 and nonlinear crystal: this distance is marked as X in our diagrams. Computations gave us a plotting for parameter \u03b4 \u2013 the graph is shown in Fig. 11. One can see from this plotting that \u03b4 takes a maximal negative value at the boundaries of stability zone. Therefore our choice for the distance was X \u224814.06 cm.<\/p>\n<\/div><div class=\"fusion-text fusion-text-10\"><p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10884\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-11.-Parameter-\u03b4-vs.-distance-X.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-11.-Parameter-\u03b4-vs.-distance-X.png\" alt=\"\" width=\"536\" height=\"334\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27536%27%20height%3D%27334%27%20viewBox%3D%270%200%20536%20334%27%3E%3Crect%20width%3D%27536%27%20height%3D%27334%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-11.-Parameter-\u03b4-vs.-distance-X-200x125.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-11.-Parameter-\u03b4-vs.-distance-X-300x187.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-11.-Parameter-\u03b4-vs.-distance-X-400x249.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-11.-Parameter-\u03b4-vs.-distance-X.png 536w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 536px) 100vw, 536px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 11. Parameter \u03b4 vs. distance X. The vertical dash lines depict the boundaries of stability zone.<\/h5>\n<p>Fig. 12 shows the cavity\u2019s stability zone in coordinates distance X and power p. One can see that at low power (the beginning of forming a Q-switch pulse), the laser operates at the boundary of stability zone, but when the Kerr lens has been launched, operation moves to a more stable mode.<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10885\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-12.-Stability-zone-for-a-resonator-grey-shading.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-12.-Stability-zone-for-a-resonator-grey-shading.png\" alt=\"\" width=\"421\" height=\"340\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27421%27%20height%3D%27340%27%20viewBox%3D%270%200%20421%20340%27%3E%3Crect%20width%3D%27421%27%20height%3D%27340%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-12.-Stability-zone-for-a-resonator-grey-shading-177x142.png 177w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-12.-Stability-zone-for-a-resonator-grey-shading-200x162.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-12.-Stability-zone-for-a-resonator-grey-shading-300x242.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-12.-Stability-zone-for-a-resonator-grey-shading-400x323.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-12.-Stability-zone-for-a-resonator-grey-shading.png 421w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 421px) 100vw, 421px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 12. Stability zone for a resonator (grey shading). The horizontal line indicates the working distance X.<\/h5>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-fusion-600 wp-image-10886\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-13.-Dependence-of-the-second-harmonic-output-power-P2\u03bd-600x393.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-13.-Dependence-of-the-second-harmonic-output-power-P2\u03bd-600x393.png\" alt=\"\" width=\"600\" height=\"393\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27600%27%20height%3D%27393%27%20viewBox%3D%270%200%20600%20393%27%3E%3Crect%20width%3D%27600%27%20height%3D%27393%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-13.-Dependence-of-the-second-harmonic-output-power-P2\u03bd-200x131.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-13.-Dependence-of-the-second-harmonic-output-power-P2\u03bd-300x197.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-13.-Dependence-of-the-second-harmonic-output-power-P2\u03bd-400x262.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-13.-Dependence-of-the-second-harmonic-output-power-P2\u03bd-600x393.png 600w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-13.-Dependence-of-the-second-harmonic-output-power-P2\u03bd.png 633w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 13. Dependence of the second-harmonic output power P2\u03bd on the DL current I for an LBO crystal at F = 2 kHz<\/h5>\n<p><img decoding=\"async\" class=\"lazyload aligncenter wp-image-10887 size-fusion-800\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-14.-Measured-autocorrelation-traces-of-locking-pulses-800x332.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-14.-Measured-autocorrelation-traces-of-locking-pulses-800x332.png\" alt=\"\" width=\"800\" height=\"332\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27800%27%20height%3D%27332%27%20viewBox%3D%270%200%20800%20332%27%3E%3Crect%20width%3D%27800%27%20height%3D%27332%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-14.-Measured-autocorrelation-traces-of-locking-pulses-200x83.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-14.-Measured-autocorrelation-traces-of-locking-pulses-300x125.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-14.-Measured-autocorrelation-traces-of-locking-pulses-400x166.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-14.-Measured-autocorrelation-traces-of-locking-pulses-600x249.png 600w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-14.-Measured-autocorrelation-traces-of-locking-pulses-768x319.png 768w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-14.-Measured-autocorrelation-traces-of-locking-pulses-800x332.png 800w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-14.-Measured-autocorrelation-traces-of-locking-pulses.png 1014w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 14. Measured autocorrelation traces of locking pulses and their sech2 fits.<\/h5>\n<\/div><div class=\"fusion-text fusion-text-11\"><p>When the optical correlatator for pulse registration on the two-photon-induced photocurrent in a GaAsP photodiode (type G1116, Hamamatsu) was applied for measurements, the duration of a single pulse from the Nd:YAG laser was 3.25 ps; this was at the average output power 1.5 W and the pulse repetition rate of Q-switch equal 2 kHz (the dependence of the second-harmonic output power P2\u03bd on the DL current I for an LBO crystal at F = 2 kHz is shown on Fig. 13).<\/p>\n<p>Fig. 14b demonstrates the measured autocorrelation function for a pulse. The scanning Fabry-Perot interferometer (parameters: free spectral range \u2013 1500 GHz, reflection coefficient of mirrors \u2013 0.96) was used for measuring the spectral bandwidth of generation \u0394\u03bd \u2248 200 GHz (see Fig. 15, where this spectral width was slightly increased due to contribution from instrumental broadening \u2248 10%). Therefore, we obtain \u0394\u03bd \u00b7 \u0394\u03c4 \u2248 0.65, which is with accuracy to factor 2 close to the case of unchirped sech2-shaped pulses.<\/p>\n<p><img decoding=\"async\" class=\"lazyload aligncenter size-full wp-image-10888\" src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-15.-The-optical-spectrum.png\" data-orig-src=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-15.-The-optical-spectrum.png\" alt=\"\" width=\"537\" height=\"393\" srcset=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27537%27%20height%3D%27393%27%20viewBox%3D%270%200%20537%20393%27%3E%3Crect%20width%3D%27537%27%20height%3D%27393%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-srcset=\"https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-15.-The-optical-spectrum-200x146.png 200w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-15.-The-optical-spectrum-300x220.png 300w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-15.-The-optical-spectrum-400x293.png 400w, https:\/\/www.szlaser.com\/wp-content\/uploads\/2022\/10\/Fig.-15.-The-optical-spectrum.png 537w\" data-sizes=\"auto\" data-orig-sizes=\"(max-width: 537px) 100vw, 537px\" \/><\/p>\n<h5 style=\"text-align: center;\">Fig. 15. The optical spectrum, measured with a scanning Fabry\u2013Perot interferometer.<\/h5>\n<p>The peak power of a separate pulse near the maximum of envelope for Q-switch pulses was \u224850 MW. In this connection, it should be noted that \u0394\u03c4 was measured from the autocorrelation\u00a0function at wavelength \u03bb = 1064 nm. When we made measurements of \u0394\u03c4 for the Q-switch\u00a0mode, it was shown that at \u03bb = 532 nm the pulse duration was approximately twice shorter.<\/p>\n<p>One might expect that this proportion in the pulse lengths will be almost the same for\u00a0operating in the QLM mode, so the actual peak power of lasing may be about \u2248 100 MW.<\/p>\n<\/div><div class=\"fusion-title title fusion-title-6 fusion-sep-none fusion-title-text fusion-title-size-three\"><h3 class=\"fusion-title-heading title-heading-left fusion-responsive-typography-calculated\" style=\"margin:0;--fontSize:24;line-height:1.3;\"><h3>Conclusion<\/h3><\/h3><\/div><div class=\"fusion-text fusion-text-12\"><p>In summary, we should note that ML mode of generation for a continuous laser using a travelling-wave AOM was obtained in papers (Kornienko et al., 1981; Kravtsov et al., 1983; Nadtocheev &amp; Nanii, 1989). The authors had noted that the band of mode locking was increased by \u2265 10 times in comparison with the case of standing-wave AOM. However, in these researches the feedback was ensured using additional mirrors within the laser cavity; this made the entire design more complicated and the Q-switch mode was absent. Our solution SMAOM, when a single AOM is enough for obtaining a stable QML mode, in combination with forming a Kerr lens, will provide high levels of pulse energy \u2248 1 mJ and peak power \u2248 50 MW at least. This design of laser does not require any additional &#8220;starting&#8221; for the making of Kerr lens and it exhibits good (long-run and short-run) stability in output characteristics and there is no need in auto-adjustment schemes.<\/p>\n<\/div><div style=\"text-align:right;\"><a class=\"fusion-button button-flat fusion-button-default-size button-default fusion-button-default button-1 fusion-button-default-span fusion-button-default-type\" target=\"_self\" href=\"https:\/\/www.szlaser.com\/index.php\/wiki\/geometric-optics\/\"><span class=\"fusion-button-text awb-button__text awb-button__text--default\">An introduction to Geometric Optics &gt;<\/span><\/a><\/div><\/div><\/div><\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":7,"featured_media":0,"parent":8230,"menu_order":143,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-10866","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v18.7 (Yoast SEO v25.8) - 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