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Download Homework Set #7 Due: 4-4-14
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Transcript
Homework Set #7 Due: 4-4-14 This homework set covers the same material as hw#6, but explores different configurations. It includes a tutorial on a vital technique used in many short pulse laser labs: OPG/OPA. (1) A beam enters a negative uniaxial crystal as shown. The crystal was cut so that the optic axis is perpendicular to the interface. If the input beam is polarized to be pure ordinary or extraordinary, it will have a well defined direction inside the crystal, shown as making an angle with respect to the optic axis. This is the angle used in the equation that determines the extraordinary index of refraction, ne(q). (a) Suppose the beam is polarized perpendicular to the page, making it an ordinary wave. Find . 45 o optic-axis no = 1.50 n = 1 45 polarization for (b) polarization for (a) (b) Suppose, instead, that the beam’s polarization is in the plane of the page making it an extraordinary wave. Find . (Rather than try and solve a complicated equation, I like to use an iterative approach here. You have the correct angle when Snell’s Law and the equation for the extraordinary index of refraction are self-consistent.) (c) Just for practice, identify if the polarization is s or p for both problems (a) and (b). Introduction to problems 2 and 3: Optical Parametric Generation, Optical Parametric Amplification The calculation I’m asking for next is fairly simple. However, this problem gives me a chance to discuss an important technique without using class time! Optical parametric generation (OPG) is a (2) process in which an input photon is split into two output photons, conserving energy. The input field is called the pump field and there are two output fields. For historical reasons, the output field with the higher frequency is labeled the “signal”, and the other one is labeled the “idler”. Since energy is conserved, we have: hp = hs + hi where p, s and i stand for pump, signal and idler respectively. As an example, a 500 nm pump field (green) could give rise to a 700 nm signal field (near IR) and a 1750 nm idler field (IR). The problem is more complicated than this, of course. Classically, an input field incident on a nonlinear crystal with a nonzero (2) can only give rise to amplification of pre-existing signal or idler fields, not actual generation. Thus, a 500 nm pump field alone would not be expected classically to generate either the 700 nm or 1750 nm fields given above. However, if a strong 500 nm pump field was accompanied by a weak 700 nm field, the 700 nm field could grow at the expense of the pump, and a 1750 idler would be generated as well. This process is called optical parametric amplification (OPA), and we have already seen this, only we called it difference frequency generation. However, a full quantum treatment reveals that OPG is possible. Even if only the 500 nm pump field were present, vacuum fluctuations would provide the initial signal or idler field. In this regard, OPA and OPG are analogous to the Raman gain and Raman generation discussed in class. n generation, the pump en nergy and thee energy spaacing betweenn the states iin the In the casse of Raman nonlinear medium deteermine the neew wavelengtths generatedd. What determ rmines whether a 500 nm pump m and 1750 nm m fields, or 800 nm and 13333 nm fieldss, or any otheer combinatioon that field geneerates 700 nm is energettically alloweed? The answ wer is that iniitially all eneergetically alllowed combinnations will ooccur, but usuallly some mod des will have higher gain than t others an and quickly coome to dominate. In partiicular, phase maatching must still be satissfied for efficcient generattion, and cann be used to select the deesired signal/idleer wavelength hs. However,, in general, OPG based oon the ampliffication of vaacuum fluctuaations followed by b competitio on between co ompeting outtput modes is unstable andd great care m must be taken to get useful outtput. One effecctive approach h is to place the t nonlinear crystal insidee an optical cavity, such ass is used for llasers. A cavity can readily be b used to prreferentially select s a givenn mode of thhe electromaggnetic field. W When OPG is done d in this fashion, f it is called opticaal parametricc oscillation ((OPO). Laserrs are often ccalled oscillatorss because of the t close anallogy between lasers and ellectrical circuuit based oscilllators. An OP PO is, in fact, a laser with the t nonlinearr crystal actin ng as the gaiin medium. OPO’s workk well, but arre not particularly relevant fo or ultrafast wo ork, at least no ot currently. However,, there is anotther approach to achieving stable outputt that is of greeat importancce. Rather thaan rely on vacuum m fluctuations to start the process, p a wh hite light conttinuum can be generated uusing (3) proccesses and this can act as the seed (initial signal s field). The T nonlinearr crystal then,, simply perfoorms OPA. Since a white ligh ht continuum provides ligh ht over a widee range of waavelengths (4000 nm to 10000 nm is comm mon), a wide ran nge of outputt fields are av vailable by ph hysically rotaating the crysttal to select w which energettically allowed fields fi will be phase p matcheed. Note that the continuum m generation need only prrovide the signal or the idler, it does not have h to proviide both. This is importannt because thee idler is ofteen too far intto the infrared to o be seeded by b continuum generation. beam spliitter (low refleectivity) mirrorr 800 nm n input B BBO crystal for SHG residual 800 nm light no longer needed dichroic beamsplitter sapphire window continuum c generation 400 nnm light (pum mp) BBO for OPA residual pump plus signal and idler beams mirror beam combinerr: 400 nm + continuum Suppose we w have a Ti:Sapphire laaser system producing p 1000 fs, 1 mJ puulses at 800 nm (commerrcially available)). While this is all well an nd good, whaat do we do iff our experim ment requires ultrafast pullses at 1200 nm or 550 nm? A possible solution s using g continuum generation sseeded OPA is indicated iin the figure abo ove. This is provided p just so s we have a simple, conccrete examplee; there are m many configuraations and this one o is not, in general, g going g to produce much m energy.. Better system ms have an addditional stagge. Here, ~300 J of 800 nm light is frequency doubled using BBO to generate ~100 J of 400 nm light. BBO has a large nonlinearity and only modest beam walk-off, so it is a good choice for this application. The 400 nm light will serve as our pump, thus allowing us to generate light throughout most of the optical range as well as in the infrared. The 400 nm light is separated from the residual 800 nm light using a dichroic beamsplitter. A small portion of the 800 nm light, of order 10 J, is also split off for continuum generation in a thin fused silica or similar window. We’ve discussed continuum generation and you know that this is primarily a (3) effect and, thus, can be excited in any medium. The 400 nm light and the white light continuum are combined in a 2nd BBO crystal where OPA takes place. Naturally, the system must be constructed so that the continuum pulses and the 400 nm pulses arrive at the same time. (The delay line to adjust the timing is not shown.) The BBO crystal converts some of the 400 nm light into signal and idler fields. The wavelengths of the signal and idler are determined by rotating the BBO crystal until an angle is found so that the desired output wavelengths are phase matched when traveling, in this case, collinearly with the pump. The pump, signal and idler can be separated using dichroic beamsplitters or other apparatus if desired. Reports of output energy for a simple system like this vary greatly, but at least several J is easily obtained. Often, a second BBO crystal is employed as an amplifier. Conversion efficiency in an optimized system can be very good, giving rise to signal pulses with ~30 J. Another way to adjust the output wavelengths is to vary the relative delay between the continuum and pump pulses. As you recall, continuum generation is accompanied with moderate chirp, thus it is possible to arrange somewhat for the pump pulse to overlap with different spectral portions of the continuum pulse. In any case, continuum generation plays a vital role stabilizing the OPG output as does a second amplifier stage. A few additional comments. Both BBO crystals must be sufficiently thin that phase matching does not constrain the output wavelengths too much. We still want the output pulses to be short, so a sufficient bandwidth must be phase matched. For BBO, crystals with a length of a few mm’s provides ample gain and yet allows the generation of short pulses. In fact, pulses shorter than the pump pulse are sometimes reported. Additional nonlinear steps involving sum- and difference-frequency generation between the residual 800 nm, pump, signal and idler fields can greatly extend the spectral range of this system. The system described here used a collinear geometry with the pump, signal and idler beams following the same path inside the crystal. Non-collinear OPA can be better. The problems. (2) What is the phase matching angle for Type I, degenerate, collinear OPG in KD*P using a 532 nm pump? “Type I” means, in this case, that the pump is extraordinary and the signal and idler are ordinary. “Degenerate” means that both the signal and idler have the same wavelength. (3) What is the phase matching angle for the OPG of 900 nm light from a 500 nm pump in KD*P? You will have to decide which beams to make ordinary and extraordinary. Some index data is provided below. ne no 500 nm 1.4697125 1.5089041 900 nm 1.4598833 1.4960746 1125 nm 1.4575388 1.4919834 Wavelength