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Mxc6L e&3!J<oY slC: /g@*o;}5g‘Vf`e.8Sq`WLb70ƻ@A|-2, f1iUlT&)O[6Za5DEJ&ޛW2X@+*T/Dt2nRٱ i&y_^im&jDIH!4X&W,/V@5bQ&x\݊nbfa?VD% (ۨuʃSUvkZ7y 5X5g\9%IG&՛T jlT%*#.J,2bM$c6% ߚi2f˜*ـJL%T _UBb 's> bmc5bzPy%vYV&a"Cb\xe eSIx"ئ{ @5bzP7-Zal-llS5brV7-#Z5k*,[[6)k)@JM))%i@SMnP5bV)%Ipo"`Ob(lfʭ&3*cLF2xY`5bVS&:oPDJh* M^Olk,K}],Ā )7-Q5Cb$0 sf,wS:m< @4̦3fC92.+#$?Lw.Q͊At48R܀5bZQe;a&Rhd@B9Ɛ&@LG n$9d"Z`.8P=nZC [`bTvWΘ4V` lkNX*6L5bRTe?Mn@U©-I%%)lXs‹lSr݅5e^HKm@d.=3KRLH%!gN}QT?)cJneQ<4VZ5V|QzWgp߶i}lۂ]YB,v@5d"Q)Sy@& .2-gZCSK<^2K_gl_ll!nvenv5gPņ͔KRctl.̾,K05 O00(̢Q+)ŠJ=r>ۀ75gt'Yϛ^݅#}iڼ\5kr'EǗ8>{vx.."jf365gd97q a+.@ko^b56"|l0MrZ/Z "%(),-./01=CDFILRTWabcdefghilmnoprstuvwy / o0 (Percent Loaded=   W{ 0 (%  *_A?` 0 (percentLoaded ?v framesLoaded_rootgetBytesLoadedtotalFramesgetBytesTotalpercentLoadedMathroundBarscale_xscalepercentloaded RX  RX  d R  O_"K T& pnBarscale  3 ( (@?<framesLoaded totalFramesI @?_root@ρ`@?D!_rootInfo_Paneltext1This animation aims to introduce an important parameter that will be used throughout the units describing laser dynamics. This parameter is the cavity decay lifetime, which as its name suggests, is a measure of how long it takes for the field inside the cavity to decay away. To simplify the analysis we shall consider the "cold" cavity only (without the laser medium). This allows us to concentrate on the output coupling loss only which is provided by one of the end mirrors as shown. text2First we shall explain the diagram opposite. You will note that we have represented the circulating field by looking at the progress of one photon around the cavity. Note that initially the field inside the cavity is some arbitrary value represented by the bar on the far right on a scale of 0 to 100% (100% representing the initial value). There are two input values that you may vary: the cavity length and the loss of the end mirror - leave them for now (you may change them later). There is also a counter counting the number of round-trips the solitary photon makes, this helps us keep track of the total TIME the process takes (we know how fast the photon is travelling and the distance it travels). Click the "tick" button on the top left to start the cavity decay.
text3The animation calculates the reduction in the intracavity field as the field completes successive roundtrips of the cavity starting from the first time the photon symbol encounters the end output-coupling mirror. The field decays away in an exponential fashion. You will note that the cavity decay lifetime is given on the upper data box (tau-cav). What is this lifetime? It is simply the time it takes the field to decay to 1/e of its original value. Observe the decay and see that this is the case. You may wish to stop the animation (cross button) and enter other loss values. We shall now proceed to derive the simple expression for the cavity decay lifetime.

text4It is now a matter of deriving a simple expression for the temporal loss of the intracavity field illustrated in the last slide. The procedure is simple, we equate the loss per round trip with the time taken for the same round trip. The loss per round trip is simply beta multiplied by the intracavity intensity. The time for one round trip is twice the cavity length divided by the speed of light c. If you wish, go back to the animation to convince yourself of this. Using these facts we are able to account for the time dependant loss of the laser cavity and arrive at the equation on the next slide.

text5This is our final expression for the optical loss of the cavity. We may now combine this with the expression derived previously for the optical gain such that our final rate equation takes shape.


text6The initial formulisation of the equation we are aiming to derive is given opposite. Note that the stimulated emission cross section that we introduced in an earlier section is an important factor here. It allows us to express the probability of stimulated emission in terms of individual atoms. The population multiplied by this probability will give us the appropriate numbers of photons produced in the bounded volume and hence the increase in intensity.

text7It is just a matter of a simple rearrangement to obtain the final expression shown on the next slide. Note that for a laser medium of refractive index n the speed of light will be given by c/n this detail is omitted from our analysis to simplify things further. The factor n is easy to introduce at a later time if the analysis requires an exact numerical answer.

text8Our examination of the optical gain is now complete. The expression for optical gain within the laser medium is just one half of the overall picture of the temporal intensity evolution.
To complete the picture we must include some loss factor into our equation. This will be the focus of the next section.

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