A Primer on Dispersion in Waveguides

Transcription

1 A Primer n Disersin in Waveguides R. S. Marjribanks 00 The linear ave equatin fr sund aves, as fr light aves, is: 1 F - F 0 [1] cs t Fr sund aves, this can be used t slve fr the scalar ressure-amlitude (x,y,z;t) f the acustic ave, r fr the scillatin-amlitude x r (x,y,z;t), r velcity-amlitude v(x,y,z;t), r hich are vectr fields. Since the scillatin and velcity can be fund frm the ressure field, e usually slve fr that. The velcity and scillatin amlitudes can be fund frm the ressure gradient, hich gives the frce term fr the equatin f mtin f the gas in hich the sund ragates. The aveguide surfaces can exert a frce erendicular t the face, but nthing in the sliding directin arallel t the surface. S the t bundary cnditins fr this secnd-rder differential equatin are that the first derivative and secnd derivative f the ressure must be zer alng the directins arallel t the surfaces. Take the directin alng the aveguide t be z, and take the transverse directins t be x ( x 5cm) and y ( y 15cm). A general lane ave can be ritten as: rt ( r r r,) ex i t -k + f [] { } Where is the frequency f the ave, and k r is the ave-vectr, having magnitude /l (here l is the avelength) and a directin erendicular t the surfaces f cnstant hase, i.e., in the directin f ragatin. The magnitude alne f the ave-vectr is knn as the avenumber. The argument ( t - k r + f) is called the ttal hase. (Nte that this is hasr ntatin.) Sh that this slutin t the ave equatin in the aveguide has the general frm: xyzt (,,,) cs( ky)cs( kcs( t - kz + f ) [3] y z z here f is arbitrary, rvided that the cnstants k x, k y, satisfy: k m and k n ; here m, n ŒZ [4] x x y The k z is then determined frm the relatin fr these erendicular cmnents: k k + k + k [5] hich becmes x y z kz ± k - Ê m n Ë Á ˆ - Ê ˆ x Ë Á [6] y Only the last term in equatin [3] evlves in time. The ttal hase, (t k z z + f ), is cnstant if z k t [7] z y DRAFT DRAFT DRAFT DRAFT DRAFT 1 lease give crrectins t marj@hysics.utrnt.ca

2 Thus e see that surfaces f cnstant hase mve dn the aveguide at seed /k z, hich is therefre called the hase seed, v f. The sitive and negative signs crresnd t aves mving in site directins. Equatin [4] ints ut that the different slutins e can have, fr a ave ragating in a aveguide, can be labelled by an rdered air (m, n) f the indices. These different ays f ragating are termed mdes f the aveguide, even as there are nrmal mdes f scillatin f a stretched ire r rectangular drum-head. It s easy t cnstruct these mdes gemetrically, fr a given avenumber k. Pragating a lane ave bliquely dn the aveguide at an arbitrary angle q frm the z-axis, reflectins at each all can be cnstructed. After reflectin frm t site-facing alls, the reflectin jins the surce ave, again ragating in the riginal directin. If the reflectin jins the surce ave n the next (r subsequent) cycle exactly in-hase, there can be cnstructive interference; any ther ssibility ill mean that multile reflectins trailing behind ill eventually cancel ut the ave entirely. Yu can icture this easily, using the abview rgram Mde Cnditins.vi n the lab PC fr this exeriment, r thrugh links n the ebsite fr the Phtnics ab, hich ermits yu t change the avenumber and angle f ragatin. Returning t ur develment, ur riginal lane ave in equatin [] ragates in the mst rdinary ay ith a hase seed equal t the free-sace ave seed c s. Putting this in the haseseed frmula, e have c s /k, hich e can ut int [6] t get: kz ± Ê m n Ë Á ˆ - Ê c Ë Á ˆ - Ê ˆ x Ë Á [8] y Fr a given mde, is there a avelength (avenumber) fr any arbitrary frequency? See that in [8], there is a minimum value f, fr any mde (m, n), and fr frequencies ler than this, k z is imaginary. The meaning f an imaginary avenumber can be seen if e g back t the fuller hasr ntatin fr [3], hich is: xyzt (,,,) cs( ky)cs( kex i( t - kz + f ) [9] { } y z z If the frequency has t l a value, it leads t an imaginary avenumber, and fr this avenumber there is n lnger a ragating slutin nly an exnentially gring slutin r an exnentially decaying slutin. Althugh bth cases are needed fr a general slutin ver a finite length, the gring slutin ill nrmally se energy rblems as it ragates ff t infinity. Therefre, in general terms there is a cutff frequency fr a given mde, and bel this frequency the slutin dies evanescently. this cutff frequency is given by: Ê ˆ ± Á + Ê ˆ c m Ë Ë Á n [10] x y (Nte that the t signs f and likeise f k z d nt cmbine t give fur slutins; nly t are unique, and crresnd t right-ging and left-ging aves. This can be seen frm the fact that cs(t k z z + f ) is an even-valued functin.) We can lt vs. k z fr different mdes, as in Fig. 1 bel: DRAFT DRAFT DRAFT DRAFT DRAFT lease give crrectins t marj@hysics.utrnt.ca

3 (rad s 1 ) Ê Ë Á ˆ + Ê ˆ c kz m Ë Á n x (0,0) (0,1) 10 4 (0,) (0,3) (1,0) (1,1) (1,) (1,3) (1,4) (1,5) (,0) (,1) (,) k z (m 1 ) Fig. 1: k disersin curve; nte mdes, labelled in rder f increasing cut-ff frequency. [needs fix] y [11] The lest mde, (0,0), ragates as des a ave in free-sace: it exhibits n disersin. The ther mdes, hever, sh different hase seeds fr different frequencies f ave. Since the hase seed is given by /k z, it is shn by the sle f a line frm the rigin t a int n any curve. S, mde (0,0) has a single, ell-defined hase seed, equal t the seed f sund in free sace. Mde (0,1) is asymttic t (0,0) at large k z, as are the ther curves, in fact, and the hase seeds all g t the same value as k gets large (shrt avelengths). But nte that as k z ges t zer (i.e., as the frequency araches the cut-ff value), the hase seed f all higher mdes ges t infinity! Behaviur f ulses Phase-frnts fr aves f different frequencies mve frard at different seeds, in this system. Of curse, a lane ave ith a ell-defined frequency exists fr all times, and cannt be turned n r ff; yu ill kn already frm Furier analysis (r frm Heisenberg s uncertainty rincile) that, fr any ave, shaing a carrier ave ith an envele that turns n r ff t make a ulse f finite duratin ill lead t nt just ne frequency but a sread f frequencies. As this ulse ragates in a articular aveguide mde, then, the varius cmnent lane aves ill ragate at different hase seeds, and change their hase relatinshi t each ther, deending n the shae f the curve in Fig. 1 fr that articular mde. Cnsequently, the ulse ill re-shae itself as it ragates, and becme smething else. Cnsider a ulse hse sectrum has a centre frequency. A simle examle uld be a symmetric ulse-amlitude envele, like a gaussian ulse, imsed n a carrier-ave f frequency. We can take the k disersin curve t give k() rather than (k) (as yu ll see if yu turn Fig. 1 n its side) and make a Taylr-series exansin arund that frequency: 1 k( ) k( ) + k ( - ) + k ( - ) [1] DRAFT DRAFT DRAFT DRAFT DRAFT 3 lease give crrectins t marj@hysics.utrnt.ca

4 here k k k k Fr a gaussian ulse, e can rite the electric field E (t) and its Furier transfrm in the time variable E ( ): -( t/ t ) it E() t e e ex t i t, E È ( - ) ( G + ) ( ) ex Í- [13] Î 4G The Furier transfrm in time is invertible it cntains all the infrmatin that E (t) des, and is an equally gd reresentatin f the electric field. S if e let the ulse ragate in z, e can d that equally ell letting the Furier transfrm evlve in z. Our Furier transfrm as in time, s the behaviur in theindeendent variable z is just as it as in equatin [], and e have the chance t recgnize that the avenumber k (smetimes called the ragatin cnstant b) deends n the frequency. Using ur Taylr-series exansin: Ez (, ) E ex [-ik z] È ik z ex -ik( ) z - ik z ( - ) - Ê 1 + ˆ [14] Í Á ( - ) Î Ë 4G This is a cmlete anser f h a ulse ill alter, ragating in a system that has disersin. It is, hever, cnvenient t cnvert this Furier transfrm back int the time-dmain t see h the avefrm in time E(z,t) ragates in sace: i t Ezt (,) Ez (, ) Ú e d - i[ t -k( ) z] e È ( - ) Ezt (,) ex - + i( - )( t - k d Ú Í Î () z - [16] - 4G 1 1 here + ik z [17] G() z G This can be integrated frmally, but it s easy enugh t d by insectin, nticing that the integral is much the same as the ne e ill have dne in [13] t get the electric field int the frequency dmain, but ith these changes G 1/[4G(], t (- ) and (t k : Ezt (,) ex [ i( t -k( ) ] ex[- G( ( t - k ] È Ê z ˆ È z exíi Át - ex ( t Ë ( ) ÎÍ - Ê Á Ë - ˆ [18] Í G Í g vf v Î here vf ( ) vg( ) k( ) 1 [19] k ( ) This e can see as a carrier ave, in the first term, and a gaussian envele in the secnd term. But n, surfaces f cnstant hase (cnstant f ttal t z/v f ) the hase frnts mve n lnger at the seed f the ave in free sace, but at the ne hase seed v f given by equatin [19]. [15] DRAFT DRAFT DRAFT DRAFT DRAFT 4 lease give crrectins t marj@hysics.utrnt.ca

5 ikeise, the ulse envele n lnger mves at the seed f the ave in free sace, but instead at the gru velcity v g, als given in [19]. Smething mre interesting still can be learned by cllecting the hase and amlitude terms searately in the exnentials in [18]: Ezt (,) ex [ i( t -k( ) ] ex[-( Re[ G( ] + iim[ G( ] ) ( t - k ] ex [ i{ ( t -k( ) -Im[ G( ] ( t - kz ) }] ex[ -Re[ G( ] ( t - kz ) ] [0] if ttal ( z, t) e ex -Re[ G( ] ( t - k [ ] The secnd term is the amlitude term, and determines the ulse shae; the first hasr term has unit amlitude and gives the field scillatins. It s argument is the ttal hase f the evlving ave. The instantaneus frequency f the ave is the rate f change f this ttal hase, hich is nt. Instead, in this case the instantaneus frequency is: (,) zt fttal(,) zt t { t -k( ) z -Im[ G( ] ( t - k } t - Im[ G( ] ( t - k S, at ne fixed sitin z in sace, as the ulse travels ast the frequency isn t cnstant it changes linearly in time. Fr ur acustic aveguide exeriment, this makes a sliding tne, like the chir f a bird s sng, hich yu can ssibly detect by ear, and certainly measure ith the equiment rvided. By analgy, this is termed a frequency-chired ulse. The value f the chir is easy t find, and deends n the gru velcity disersin (GVD), given by k ( ) ik z [] G() z G Write G( a( i b( in its real and imaginary arts; then ratinalize the denminatr: 1 1 az () bz () + i [3] G( a() z - ib() z a () z + b () z a () z + b () z Equating the real and imaginary arts f [] and [3], e can identify: G az () 1 + ( kz G ) [4] kz G 1 bz () 1 + kz G 1 + 1/ kz G S the instantaneus frequency is given by ( t - k (,) zt / kz G And frm the ther term f equatin [], the half-idth f the ulse, measured at 1/e f the maximum, is: [1] [5] DRAFT DRAFT DRAFT DRAFT DRAFT 5 lease give crrectins t marj@hysics.utrnt.ca

6 t( 1 az () The FWHM can then be calculated: 1 + kz G G + kz t( ln 1 G [7] G This is very interesting: fr this gaussian ulse, the ulseidth n deends n the distance z ragated. The ulse starts ith its riginal duratin, but then as it ragates at the gru velcity it stretches ut further and further. Fr an animated illustratin f this, see the demnstratin n the lab ebage (TravelChir.mv). [6] In summary, nte that k( ) determines the hase velcity, k ( ) determines the gru velcity, and k ( ) determines the sreading f the aveacket, the gru velcity disersin. k k( ) vf ( ) hase velcity dk 1 1 k [8] d v gru velcity g( ) dk d Ê 1 ˆ k gru velcity disersin d d Á Ë vg( ) DRAFT DRAFT DRAFT DRAFT DRAFT 6 lease give crrectins t marj@hysics.utrnt.ca