Maximizing Energy in Terahertz Pulse Radiation from a Switched Oscillator

Transcription

1 Sensor and Simulation Notes Note 554 July 1 Maximizing Energy in Teraertz Pulse Radiation from a Swited Osillator Carl E. Baum and Prasant Kumar University of New Mexio Department of Eletrial and Computer Engineering Albuquerque NM Abstrat For a THz pulse (damped sinusoid) radiator onsisting of a dipole-like antenna separated by a dieletri from a ground plane tere are many design onsiderations. Tese inlude te stored energy radiation effiieny and resistive losses. 1

2 1 Introdution Pursuing te design of a pulsed (damped sinusoid) THz radiator we need to find some optimization onditions. Given some frequeny f ow mu energy an we radiate in a damped sinusoidal pulse? Tis depends on te various parameters of te antenna and te soure. For present purposes let us onsider te onfiguration in Fig (a) Side view (b) Front view Figure 1.1: THz radiator. Here we ave a onduting ground plane (typially opper). Tis is separated from te antenna by a dieletri of permittivity ɛ and tikness. Te antenna is a alf-wavelengt radiator of lengt l a. Ea alf is arged to ±V. Tese alves are separated by a swit medium (say semi-insulating gallium arsenide) of lengt l s. Te widt of te antenna is w. Te swit is illuminated by a femto seond laser to ause it to ondut (lose).

3 Maximizing te Stored Energy Te stored energy is U = 1 C[V ] = CV. (.1) Te energy available at f te dominant alf-wave resonant frequeny is estimated as [1] U 1 = 8 π U.81U (.) te remaining energy being for iger order resonant frequenies wi we neglet. We need to maximize tis energy and radiate as mu of tis as possible. Tus we need to maximize CV. Consider te apaitane C ɛwl a /. For a given tis is maximized by large w te lengt l a being given by λ os l a = v f v ɛ 1/ r ɛ r = ɛ ɛ or more aurately > v > ɛ 1/ r ( os = osillator) (.3) (.4) wit v approaing ɛ 1/ r for large w sine most of te apaitive energy lies between te antenna and te ground plane. However we do not want w to be so large tat iger order osillation modes are supported. So we migt limit te widt as w l a l s for obtaining a large apaitane. Tis apaitane an be estimated as C ɛw[l a l s ] 4 as two apaitors in series. One an maximize C by minimizing but it is te energy we wis to maximize. So onsider maximizing V. As an approximation we an estimate tis via (.5) (.6) V 1 = E d E d average breakdown eletri field troug dieletri to ground plane. (.7) Tis inludes te edge effets on te antenna wi an be mitigated to some extent by roll-ups on te edges. Anoter voltage limitation onerns te swit wi needs to be igly insulating before te arrival of te fs laser pulse. Let us estimate te voltage stand-off as V = E s l s E s average breakdown eletri field troug swit between two antenna alves. (.8) 3

4 As we inrease V and aordingly inrease we ave U = CV = ɛw[l a l s ] [E d ] = ɛw[l a l s ] E d (.9) tereby inreasing U. Similarly as we inrease V and aordingly inrease l s we ave U = CV = ɛw[l a l s ] [E s l s ]. (.1) Inreasing ten (.9) inreases te stored energy up until te swit limits te voltage. Inreasing beyond tis dereases te stored energy. Equating te two results gives ɛw[l a l s ] Ed = ɛw[l a l s ] [E s l s ] = E s (.11) l s E d as an approximate optimum. However wile one may wis λ 4 (.1) for radiation arateristis we need l s < l a (.13) as a limiting fator wi may alter te results of (.11). For example if E d > E s (.14) ten our limitation on te energy is given by (.1) wit te limit in (.1). In tat ase we need to maximize (for fixed l a ) X = [l a l s ]l s. (.15) Differentiating wit respet to l s gives = l a l s 3l s l s l a = 3. (.16) Tere are still oter fators to onsider su as swit losses and ondutor losses. 4

5 3 Ideal Radiation Carateristis 3.1 Single eletri dipole radiator Let us make some simple estimate of te radiated fields. An eletri dipole in free spae as far fields [] E f ( r s) = µ 4πr s e γr 1 r [ 1 r p(s)] = µ s 4πr 1 r p(s)e γr (3.1) H f ( r s) = Z 1 1 r E f ( r s) Laplae transform (two sided) γ s/ propagation onstant s Ω + jω Laplae-transform variable or omplex frequeny Z [µ /ɛ ] 1/ wave impedane of free spae p(s) p(s) 1 z eletri dipole moment 1 r 1 1 r 1 r = 1 θ 1 θ 1 φ 1 φ transverse dyad. Tis gives a radiated power s = jω as P 1 = 1 [ Ef ( r jω) Hf ( r jω)]4πr ds (3.) S S spere of radius r. Te fator 1/ gives te average power over a yle. Appropriately substituting gives P 1 = µ ω 4 p(jω) 1 3π r r p( jω) 4πr ds S = µ ω 4 π π p(jω) 1 z 1 r 1 z sin(θ)dφdθ 8π = µ ω 4 π 4 p(jω) [ 1 z 1 θ ] sin(θ)dθ (3.3) = µ ω 4 π 4 p(jω) sin 3 (θ)dθ = µ ω 4 3 p(jω).33µ ω 4 p(jω) = [µ ɛ ] 1/ speed of ligt. 3. Two dipole radiators λ/ apart Modifying tis for our geometry let us approximate tis as two dipoles separated by a alfwavelengt 18 out of pase but only radiating into a alf-spae. Let one dipole be at y = +λ/4 5

6 wit te seond at y = λ/4. Ten we an superimpose te two dipole fields as E f ( r s) = µ 4πr s 1 r p(s) 1 z [e γ[r λ 1 4 y 1 r] e γ[r+ λ 1 4 y 1 r] ] = µ ( ) γλ 4πr p(s) 1 r p(s)e γr sin sin(θ) sin(φ) (3.4) 4 H f ( r s) = Z 1 1 r Ef ( r s). Tis is an odd funtion of φ (or sin(φ)). For s = jω we also ave γ = jk = j ω. (3.5) Now it is onvenient to ange variables so tat te argument of sin is a funtion of only one oordinate. For tis purpose oose (x y z ) = (z x y) ( 1 x 1 y 1 z ) = ( 1 z 1 x 1 y ) Tis implies r = [x + y + z ] 1/ = [x + y + z ] 1/ z = r os θ ψ = [x + y ] 1/ = r sin(θ ) x = ψ os(φ ) = r sin(θ ) os(φ ) (3.6) y = ψ sin(φ ) = r sin(θ ) sin(φ ) 1 r = 1 z os θ + 1 ψ sin(θ ) = 1 z os(θ ) + 1 x sin(θ ) os(φ ) + 1 y sin(θ ) sin(φ ) = 1 z os(θ) + 1 ψ sin(θ) = 1 z os(θ) + 1 x sin(θ) os(φ) + 1 y sin(θ) sin(φ). sin(θ) sin(φ) = os(θ ) 1 x 1 r 1 x = 1 [ 1 x 1 r ] = 1 sin (θ ) os (φ ). (3.7) Ten we ave te radiated power P = 1 [ Ef ( r jω) Hf ( r jω)] 1 r ds (3.8) S = µ ω 4 3π r p(jω) 1 x ( ) kλ 1 r 1 x 4 sin 4 os θ ds = µ ω 4 π p(jω) = µ ω 4 p(jω) S π π π sin ( kλ 4 os θ ) [1 sin (θ ) os (φ ) ] sin(θ )dφ dθ sin ( kλ 4 os θ ) [ sin (θ ) ] sin(θ )dθ. 6

7 Substituting gives η = os(θ ) dη = sin(θ )dθ (3.9) sin (θ ) = 1 os (θ ) = 1 η P = µ ω 4 Substituting gives p(jω) 1 ν = kλη 4 dν = kλ 4 dη P = µ ω 4 [ kλ 4 4 p(jω) sin (ν) 1 + kλ sin ( kλ 4 η ) [1 + η ]dη. (3.1) [ ] ] 4ν kλ (3.11) dν (3.1) wi is simplified using Matematia 1 as [ P = µ ( ω 4 4kλ os kλ ) p(jω) (k λ 4) sin ( ) ] kλ +. (3.13) k 3 λ 3 3 For a resonant (alf-wave) ondition we ave λ = l a f λ = f = ω π = λ = k λ = ω λ = π. In (3.13) we ave ( ) ( ) kλ kλ sin = os = 1 giving P = µ ω 4 p(jω) l a (3.14) [ 4 k λ + ] = µ [ ω p(jω) π + ].38 µ ω 4 p(jω). (3.15) 3 Noting tat wit te ground plane te radiation is only in te z diretion we ave P 3 = P.19µ ω 4 p(jω) (3.16) as te forward radiated power. 1 ttp:// 7

8 3.3 Appliability Te foregoing analysis applies stritly to an imaged dipole in free spae. Te permittivity of te medium above te ground plane alters tese results somewat. So tey sould be applied for relatively small ɛ r. As ɛ r is inreased te alf-wave resonant lengt beomes less tan λ/ in free spae making te antenna a less effiient radiator. It sill radiates but it takes a longer time to radiate te energy U 1 giving a iger Q. Also as ɛ r is inreased tere an be surfae bound waves propagating along te dieletri surfae. Tis takes some of te energy in an undesirable diretion. Even for ɛ = ɛ te antenna is no longer eletrially small making te dipole araterization of te antenna only approximate. 3.4 Two losely spaed eletri-dipole radiators Now let te two dipoles be plaed at z = ± λ 4. (3.17) Ten (3.4) is replaed by E f ( r s) = µ 4πr s p(s) 1 r 1 z e γr sin(γd sin(θ) sin(φ)) µ 4πr s p(s) 1 r 1 z e γr γd sin(θ) sin(φ) (3.18) H f ( r s) = Z 1 1 r Ef ( r s). Making te same oordinate transformation as in (3.6) and (3.7) gives a radiated power P = 1 [ Ef ( r jω) Hf ( r jω)] 1 r ds s = µ ω 4 8π r p(jω) 1 x 1 r 1 x [k] os (θ ) ds (3.19) = µ ω 4 π p(jω) [k] = µ ω 4 p(jω) [k] Substituting as in (3.9) gives S π π π P = µ ω 4 1 p(jω) [k] η [1 + η ] dη = µ [ ω 4 1 p(jω) [k] ] 5 = 8 [ ] µ ω 4 ω p(jω).53 µ ω 4 15 os (θ )[1 sin (θ ) os (φ)] sin(θ ) dφ dθ os (θ )[ sin (θ )] sin(θ ) dθ. [ ] ω p(jω). 8 (3.) (3.1)

9 Note te extra fator of [k] lowering te radiated power. Tis fator is te square of te fration of a alf wavelengt given by l a. Te eletrially small antenna is a less effiient radiator. Noting tat te radiation in only in te +z diretion we ave P 3 = P = 4 [ ] µ ω 4 ω p(jω).7 µ [ ] ω 4 ω p(jω). (3.) 15 4 Mating to Eletri Dipoles 4.1 Dipole Carateristis Our formulae are in terms of te dipole moment at te resonant frequeny. We need to evaluate p(jω). Wen te swit is losed (ideally instantaneously) tere is generated a square wave osillation wit frequeny f. Te urrent in tis wave is I = V Z Z Z w w Z w Z ɛ 1/ r = transmission line arateristi impedane (4.1) [ ] 1/ µ wave impedane. (4.) ɛ Tis is only approximate. Z is lowered by fringe fields if w is not. Furtermore te impedane is influened by ɛ above te dieletri as well as ɛ in te dieletri. In [1] it is sown tat te peak urrent in te resonant mode is I max = 4 π I. (4.3) In tis resonant mode te urrent is I max at te antenna enter but zero at te ends giving ( ) πz I(z) = I max os (4.4) l a as te spatial distribution of te urrent. An alternate formula [] (ompared to arge times distane) for an eletri dipole moment is p(jω ) = 1 s la/ l a/ Ĩ(z s) dz = I max jω la/ l a/ ( πz os l a ) dz = I max jω ( ) la π = j 8 l a I π = j 8 l a V j 8 l a w V j 4 λ w V = j 8 ω π ω Z π ω Z w π ω Z w π ω w V. Z w Hene we see te advantage of a large (but not too large) widt w. As disussed in Setion we may not use te λ/4 for but someting less to inrease te stored energy as in (.1). Tis may also inrease te Q of te resonane exept for possibly oter losses. (4.5) 9

10 4. Radiation Q Te radiation is of ourse in te form of a damped sinusoid. Te Q of tis resonane an be estimated as Q = πn N = number of yled for field to fall to e 1 or energy to e. In one yle te energy radiated is U 3 P 3 T T = 1 λ f l a = period of resonane. Te frational energy radiated in one yle is U = U 3 = π P 3 T. U 1 8 U Te frational field lost in one yle is F U = π 16 giving P 3 T U (4.6) (4.7) (4.8) (4.9) N = 1 F = U = 16 U π P 3 T (4.1) Q = πn = 16 U π P 3 T. 4.3 Combined results for = λ/4 Combining (4.5) wit (3.16) gives P 3 = 1 [ ] [ µ ω 4 8 π π V ω Z w ] 1.5 V Z w. (4.11) Tis result is for = λ/4 wi is also of te order of w. So radiated power is of te general order of V /Z w as one migt expet. Assuming te dieletri as te basi limitation wit V E d U ɛw[l a l s ] V. te Q is ten Q = πn 16 π ɛw[l a l s ] V Z w 1.5V (4.1) (4.13) w[l a l s ]. (4.14) l a l a Tis is of te general order of. (sine it is assumed tat l a l s and w ; Q w/ =.) indiating a igly damped osillation for wi our approximations are inaurate. Tis low Q is assoiated wit our very fat dipoles. Tis would ange signifiantly if w were. 1

11 4.4 Combined results for λ/4 Using te results of (3.) instead we ave P 3 = 4 µ ω 4 15 [ ω ] [ ] 8 V = 56 π ω Z w 15π [ ω ] V Z w. (4.15) Now we see te power redution by te fator [ω /]. We see a power radiated proportional to V and a field proportional to V. Again in energy as in (4.1) we ave Q = πn 16 π ɛw[l a l s ] [ ] V 15π Z w 56 ω V 15π [ ] w[l a l s ]. (4.16) l a 3 l a ω Wit l a / we migt make l s small sine te swit needs to old off less voltage. Tis gives Wit Q 15π 3 [ ] w = 15 [ ] w = 15 w ω 18π f 18π [ ] λ = 15 w 3π [ ] la. (4.17) w l a tis beomes Q 15 64π [ ] 3 la (4.18) (4.19) allowing one to adjust to some desired Q. 4.5 Limitations Te foregoing is for te ase of air dieletri between te antenna and te ground plane. Pratially one needs a dieletri plane wit relative dieletri onstant of a few to support te antenna. Tis will sorten l a for a given resonane frequeny tereby lowering te effiieny of radiating into air tereby raising te Q. Tere may also be some tin dieletri to oat te swit and peraps antenna to inrease te voltage standoff to > V as desired. 5 Effet of Dieletris Of neessity tere needs to be a substrate to support te antenna above te ground plane as in Fig Tis ompliates te eletromagneti analysis introduing surfae waves guided by te dieletri over te ground plane. Tese surfae waves ave been studied by [3 6]. It is noted tat te utoff effiieny is obtained just below te utoff tikness of te H surfae-wave mode. Te E surfae wave mode does not ave a utoff frequeny but for small is only weakly exited by te antenna. Maximum effiieny 11

12 ours just before te H surfae wave mode an be exited. Te optimum substrate tikness satisfies [7] opt λ 4 [ɛ r 1] 1/. (5.1) Furtermore it would appear tat te presene of a superstrate may furter improve matters [7]. Note tat te dieletris derease te size of te antenna for te resonant ondition. Tis will signifiantly affet te Q. 6 Skin-Effet Losses Tere are losses assoiated wit te finite ondutivity of te metal. Te skin dept for a good ondutor is [8] [ ] 1/ [ ] 1/ 1 δ s = =. (6.1) ωµ σ πfµ σ Tis gives a surfae impedane [ Z s = R s + sl s = [1 + j][σδ s ] 1 ωµ ] 1/ = [1 + j] (6.) σ [ ωµ ] [ ] 1/ 1 7 1/ f R s = = π surfae resistane. σ σ At a frequeny f of.3 THz we ave for opper R s =.14 Ω. To model tis loss onsider te antenna over te ground plane as a transmission line wit [ ] sl Z + R 1/ (s) = arateristi impedane sc (6.3) γ(s) = [[sl + R s ]sc ] 1/ propagation onstant (6.4) L µ indutane per unit lengt w C w ɛ ɛ r apaitane per unit lengt R R s resistane per unit lengt. w Tere is also a small orretion to L from L s but we neglet tis. Note te R s to aount for losses in bot te antenna and te ground plane. As a transmission line it propagates a wave wit propagation approximated as [ ] 1/ γ(s)z = s[l C ] 1/ 1 + R z sl = s ] [1 + R v sl + O(s ) z (6.5) = s v z + R z + O(s 1 ) Z 1

13 for ig frequenies. Te first term is delay and te seond term is loss wit [ ] L Z = = Z C w w [ µ ] 1/ Z w = = ɛ 1/ r Z. ɛ Te propagating wave goes like e γ(s) exp [ s ] ] v z exp [ R z Z (6.6) (6.7) = exp [ s ] [ v z exp R ] s z. (6.8) Z w One period of osillation orresponds to a round trip of distane l a for wi [ exp R ] s z 1 R s z Z w Z w 1 l a R s for l a R s 1. (6.9) Z w Z w So tat l a R s Z w is te frational loss of field amplitude. Te number of yles to e 1 is ten N = Z w = ɛ 1/ Z r = ɛ 1/ Z r l a R s l a R s λ R s (6.1) Q = πn = πɛ 1/ Z r. λ R s (6.11) Note tat te swit lengt l s is assumed negligible in tis alulation. As an example te Q as a funtion of /λ and ɛ r for f =.3 THz (R s =.14 Ω) is sown in Fig From tis we an see tat te opper skin losses do not appear to be a problem as long as /λ is not too small. (a) ɛ r sweep. (b) Density plot of Q. Figure 6.1: Quality fator (Q) as a funtion of /λ and ɛ r at f =.3 THz as per (6.11). 13

14 7 Swit Losses Te swit (1.1) as lengt l s widt w and some tikness to be determined ( ). Sine we want te swit to ondut for a long time after te fs laser illumination a good swit material is SI-GaAs (Cr doped) wit arateristis [9] 5 1 ps = arrier lifetime.1 m = mobility (7.1) V s 1 4 Ω m = resistivity 5 M V = breakdown field. m Te swit resistane depends on te arrier density generated by te fs laser. Compared to te skin-effet resistane te swit resistane an be somewat larger and still aieve an aeptable Q. Immediately we an tat te arrier lifetime an be a limitation depending on te antenna Q. At.3 THz te period is 3.3 ps wi gives 3 yles in 1 ps orresponding to a Q of about 94. So one may wis to oose to get te number of yles down to 3 or so (large allows larger V ). Assume as in Fig. 1.1 tat l s is small and an be modelled after losure by a resistane R sw (time independent). As sown in Fig. 7.1 fold te antenna and ground plane. Figure 7.1: Equivalent transmission-line model of antenna wit swit. Here te refletion oeffiient at R sw is ρ = Z R sw Z + R sw = 1 R sw Z 1 + R sw Z 1 R sw Z for small R sw Z. (7.) In [1] te number of yles to e 1 is N = 1 ln(ρ) Z R sw Q = πn π R sw Z. (7.3) 14

15 How small sould R sw be? Coose te parameters Z Z w w < λ in dieletri (7.4) 4 Z w Z ɛ 1/ r ɛ r 3. Let w λ 4 λ 8 w 1 Z Z w = Ω. For N = 3 we ave R sw = Z N = Z w 4N (7.5) 1.8 Ω. (7.6) Tis an be used to estimate te SI-GaAs doping and fs laser parameters. 8 Conluding Remarks A THz pulse (damped sinusoid) radiator as in Fig. 1.1 as many design optimization questions. One needs to maximize te stored energy depending on te dieletri tikness and swit dimensions. Te antenna needs to radiate most of tis energy and tere is a tradeoff between amplitude and number of yles (Q). Te skin-effet losses are not signifiant but te swit losses need to be quantified and made aeptably small. As usual one an estimate te ombined effet of te various previously disussed fators via Q 1 = n N 1 = n Q 1 n (quality fator) (8.1) N 1 n (number of yles to e 1 ). Referenes [1] C. E. Baum Combined Eletri and Magneti Dipoles for Mesoband Radiation Part. Sensor and Simulation Note 531 May 8. [] C. E. Baum Some Carateristis of Eletri and Magneti Dipole Antennas for Radiating Transient Pulses. Sensor and Simulation Note 15 Jan

16 [3] P. B. Katei and N. G. Alexopoulis On te Effet of Substrate Tikness and Permittivity on Printed-Ciruit Dipole Properties. IEEE Trans. AP-31 pp [4] N. G. Alexopoulis P. B. Katei and D. B. Rutledge Substrate Optimization for Integrated- Ciruit Antennas. IEEE Trans. MTT-31 pp [5] D. M. Pozar Considerations for Millimeter-Wave Printed Antennas. IEEE Trans. AP-31 pp [6] F. Swiring and A. A. Oliner Millimeter-Wave Antennas. C. 17 in Y. T. Lo and S. W. Lee (eds.) Antenna Handbook VanNostrand Reinold [7] N. G. Alexopoulis and D. R. Jakson Fundamental Superstrate (Cover) Effets on Printed- Ciruit Antennas. IEEE Trans. AP-3 pp [8] C. E. Baum Teraertz Antennas and Osillators Inluding Skin-Effet Losses. Sensor and Simulation Note 535 Sept. 8. [9] K. Sakai Teraertz Optoeletronis. Springer Heidelberg 5. [1] C. E. Baum Swited Osillators. Ciruit and Eletromagneti System Design Note 45 Sept.. 16