IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, …

1 ieee TRANSACTIONS ON APPLIED superconductivity , VOL. 19, NO. 4, AUGUST 20093653A Superconductor THz Modulator Basedon Vortex Flux FlowFarrokh Sarreshtedari, Mehdi Hosseini, Hamid Reza Chalabi, Ali Moftakharzadeh, Hesam Zandi,Sina Khorasani, Member, ieee , and Mehdi Fardmanesh, Senior Member, IEEEA bstract A terahertz modulator based on the Type-II super-conductor flux flow oscillator has been proposed. Analytical calcu-lations are presented and the effects of intrinsic and extrinsic pa-rameters such as disorder strength of crystal, penetration depth,frequency, and amplitude of the modulated current on the radi-ation power spectrum have been studied. The proposed structurealso exhibits a mixer-like behavior, in the sense that its output har-monics range from the washboard frequency up to the supercon-ductor gap frequency, so the input signal is practically mixed withthe washboard frequency and its harmonics. The modulation indexfor each harmonic of this modulator has also been well-featured modulator has a potential to be used in next-generation terahertz integrated Terms Mixer, modulator, superconducting devices,vortex flux INTRODUCTIONSPECTRUM of electromagnetic radiation from a type-IIflux flow oscillator (FFO) has been derived previously byBulaevskii and Chudnovsky [1].

2 Using an analytical approachand considering assumptions such as laminar flow of vortices[2], ignoring surface Meissner current and the back effect ofthe radiation on lattice motion, they have found the radiationpower for different lattice parameters. It is shown that for atype-II superconductor in vortex flux flow regime, as vorticescome to the surface periodically with the frequency, thesystem radiates with harmonics of this frequency (washboardfrequency) up to the superconducting gap. References [3]and [4] have studied the effect of current modulation on themotion characteristics of moving vortex lattice and washboardfrequency for specific materials. In our work, based on thelinear relation betweenand, which is satisfied in a pin-freeapproximation (when the Lorentz force is so strong that itexceeds the pinning force [5]), the effect of current modu-lation on the spectrum of the radiation power is analyticallyderived.

3 We have considered the case of square vortex latticein a type-II isotropic superconductor and the dependence ofoutput frequency spectrum and modulation index on differentsystem parameters are investigated. The proposed device hasthe potential ability to be used as the mixer/modulator block ofsubmillimeter integrated receivers, where FFO devices are usedManuscript received November 26, 2008; revised December 24, 2008. Firstpublished April 17, 2009; current version published July 29, 2009. This paperwas recommended by Associate Editor N. F. authors are with the Superconductive Electronics Research Laboratory,School of Electrical Engineering, Sharif University of Technology, Tehran,Iran(e-mail: versions of one or more of the figures in this paper are available onlineat Object Identifier 1. Schematic of the the base device. Other technologies of existing mixers/mod-ulators include superconductor insulator superconductordevices, Schottky Barrier diodes, hot electron bolometers,and metamaterials [6] [9].)

4 The frequency spectrum in theoutput of this device, like all nonlinear mixers, consists ofsum and difference harmonics of its input signals and the basefrequency (washboard frequency). When all harmonics, expectthe lowest order mixing products are filtered out, it may also beexploited as a terahertz (THz) amplitude modulator, since theinput signal modulates the washboard frequency as the 1 shows a typical mixer/modulator and the correspondinginput and output signals. As it is shown in Fig. 1, the carrierfrequency (which is in fact fed from the washboard oscillation)is determined by external bias parameters of the system, suchas perpendicular magnetic field, bias current, and char-acteristic parameters of the superconductor material a vortex lattice moving at velocityand lattice constant, the carrier frequency is given by; however, dueto the nonlinear nature of FFO radiations, harmonics of this fre-quency exist up to the superconductor gap frequency.

5 We alsohave calculated the depth of modulation in the AM modulatorconfiguration, which is the ratio of the modulated componentamplitude to the amplitude of the carrier. For simplicity, we haveused the geometry and notations of [1] in derivation of THEORETICALMODELFor studying the electromagnetic radiation from a type-II su-perconductor, the geometry of Fig. 2 is chosen. The supercon-ductor is located within, and the modulatedradiation power is then found forand. Theapplied magnetic fieldalong the z-axis is assumed to satisfy, withandbeing the first and thesecond critical fields of the type-II using a quasi-static approximation and Ginzburg Landaurelations in the presence of magnetic field, one can find the fields1051-8223/$ 2009 IEEEA uthorized licensed use limited to: Sharif University of Technology. Downloaded on August 4, 2009 at 02:28 from ieee Xplore. Restrictions apply. 3654 ieee TRANSACTIONS ON APPLIED superconductivity , VOL.

6 19, NO. 4, AUGUST 2009 Fig. 2. Type-II superconductor with , magnetic field , andthe moving vortex lattice with velocity .inside the superconductor through solving the following equa-tions:(1)(2)whereis the unit vector along the z-axis,is the speedof light,andare, respectively, the magnetic and electricfields,is the unit step function,is the London penetra-tion length, andis the position of the center of thethvortex line in the vortex lattice, wherewithandbeing integers. Referring to Fig. 2 and assuming a modulatedcurrent, the velocity of a specific vortexis denoted by(3)By integration, we obtain the coordinates as(4)(5)Here,accounts for disorder in the position of vorticesalong y-axis. We now proceed to evaluate the electromagneticfieldsand. The solution of (1) can be constructed byfinding the homogenous solutionand the particularsolutioncorresponding to the inhomogeneous , the associated electric fieldsandwill be ob-tained by (2)(6)The homogenous parts ofandare determined by thecontinuity of the fields at the boundaries of the superconductor[1], [10].

7 Using (3) and assuming, the step function can beapproximated by(7)Simplifying the expressionin (1) yields(8)Using (8), (1), and identity(9)can be found as(10)Authorized licensed use limited to: Sharif University of Technology. Downloaded on August 4, 2009 at 02:28 from ieee Xplore. Restrictions apply. SARRESHTEDARIet al.: SUPERCONDUCTOR THz MODULATOR BASED ON VORTEX FLUX FLOW3655 And after some simplifications, we find the expression formagnetic flux density as(11)By integration overcomponent of the wave vectorandconsidering a specific harmonic with, we get(12)Using the same approach adopted in [1], the parameterde-fined as the ratio of electrical to magnetic field, is given by(13)Using the relation, Maxwell equations, and (6), wecan express the total fieldonly byandas(14)Evaluating the radiation power at frequency, outside the su-perconductor onas in Fig. 1, results in(15)It should be noted that there is a small difference betweenright-propagating and left-propagating radiation powers, whichis negligible [1].

8 Ignoring the relatively small terms and simpli-fications gives(16)The power radiation can be equivalently rewritten as(17)where, and the structural factoris givenby(18)III. RESULTS ANDDISCUSSIONSWe have obtained the frequency contents of the modulatorthrough approximation of the radiated power (17) for differentlimits as introduced in [1]. Different cases are investigated inthree sections, according to the magnitude of dimensionlesstransversaland longitudinalcorrelation lengths formoving vortex lattice which are expressed in units of theintervortex spacing a [1] and [11]. These three approximationregions are considered as weak disorder and large crystal, weakdisorder and small crystal, and strong disorder. All calculationsare based on the numerical data given in Table Weak Disorder and Large CrystalFor the case of weak disorder, we assumeand neglect the variation of denominator with respectto. By using the identities(19)(20)Authorized licensed use limited to: Sharif University of Technology.

9 Downloaded on August 4, 2009 at 02:28 from ieee Xplore. Restrictions apply. 3656 ieee TRANSACTIONS ON APPLIED superconductivity , VOL. 19, NO. 4, AUGUST 2009 TABLE INUMERICALVALUES OFMODULATORFig. 3. Normalized power versus angular modulator radiation power spectrum is found as(21)In Fig. 3, we have shown the power of harmonics versus an-gular frequency of each harmonic order. It can be inferred thatthe carrier component has a constant power versus frequencyand the modulated components show a parabolic behavior, asgiven by (21).B. Weak Disorder and Small CrystalAssuming weak disorderand small crystal di-mensions, the radiation power can be found as(22)Fig. 4. Modulation index versus angular frequency for different carrier 5. Schematic of normalized power of different carrier frequencies togetherwith their corresponding Strong DisorderFinally, in the case of stronger disorderwith smaller correlation lengths, the radiation power is sim-plified as(23)It can be inferred from (22) and (23) that the behavior of thesetwo cases is roughly the same, but the amplitudes would be dif-ferent.

10 In Fig. 4, the frequency contents of the modulator forcases B and C is shown for the first ten harmonics of a typicalhigh-TC superconductor for the proposed modulator. It is ev-ident in Fig. 4 that the power increases quadratically with theharmonic order number. Although the variation of frequencycontents of modulator for case A is essentially different fromthe other, the modulation index was observed to be the same, asillustrated in Fig. licensed use limited to: Sharif University of Technology. Downloaded on August 4, 2009 at 02:28 from ieee Xplore. Restrictions apply. SARRESHTEDARIet al.: SUPERCONDUCTOR THz MODULATOR BASED ON VORTEX FLUX FLOW3657IV. SUMMARY ANDCONCLUSIONSWe have shown wideband THz mixing and modulation ef-fects, through modulating the bias current of a type-II super-conductor FFO by a small ac signal. Calculations reveal thatthe radiated power from vortex flow has two modulated compo-nents accompanied by harmonics of washboard frequency up tothe superconducting changing either the bias current, or the perpendicular mag-netic field, one can tune the washboard frequency of movingvortex lattice, and therefore, the carrier frequency of proposedmodulator.