Comparative Study Between Dispersive and Non-Dispersive Dielectric Permittivity in Spectral Remittances of Chiral Sculptured Zirconia Thin Films

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1 Comparatve Study Between Dspersve and Non-Dspersve Delectrc Permttvty n Spectral emttances of Chral Sculptured Zrcona Thn Flms Ferydon Babae * and Had Savalon 2 Department of Physcs Unversty of Qom Qom Iran. 2 Department of Physcs Unversty of Tehran North-Kargar Street Tehran Iran. * Correspondng author: Tel: ; Fax: ; Emal: fbabae@qom.ac.r Abstract The transmsson and reflecton spectra from a rght-handed chral sculptured zrcona thn flm are calculated usng the pecewse homogenety approxmaton method and the Bruggeman homogenzaton formalsm by consderng that the propagaton of both dspersve and non-dspersve delectrc functon occurs for axal and non-axal states. The comparson of spectral results shows that the dsperson of the delectrc functon has a consderable effect on the results. In axal exctaton of cross-polarzed reflectances and co-polarzed transmttances the dsperson effect becomes more pronounced at wavelengths further away from the homogenzaton wavelength. Ths s also true n case of non-axal exctaton of crcular transmttances whle there are consderable dfferences for cross-polarzed reflectances where wavelength the frst Bragg peak occurs. At wavelengths n the vcnty of the homogenzaton wavelength the dsperson effect of the delectrc functon n becomes more sgnfcant. Keywords: Chral sculptured thn flms; Bruggeman formalsm; Pecewse homogenety approxmaton method. Introducton In order to obtan the transmsson and reflecton spectra for sculptured thn flms usually the relatve delectrc constant of the flm materal at a certan frequency s consdered then the relatve permttvty scalars are estmated usng the Bruggeman

2 homogenzaton formalsm. These scalar quanttes are assumed constant n the procedure of obtanng the transmsson and reflecton spectra at all frequences [-3]. However we know that the relatve delectrc constant vares wth the frequency. In addton n the man body of the lterature on the delectrc dsperson functon effect on remttances reflecton and transmsson of sculptured thn flms the smple sngle-resonance Lorentzan model s used [4-6] whle n order to be able to defne the oscllator strengths resonance wavelengths and absorpton lnewdths one should have a good knowledge of the oscllatory and quantum behavor of the thn flm. In addton there s an argument that all thn flms may not obey from the sngleresonance Lorentzan model and are composed of a double-resonance [7] or multpleresonance [89] systems. In order to avod such complcatons we have mplemented the expermental results of the refractve ndex of zrcona thn flm at each wavelength to calculate the dsperson of the delectrc constant. It should also be mentoned here that the refractve ndex for zrcona n the wavelength regon examned n ths work has only real values and the magnary part s zero []. In our earler works we have reported the reflectance and transmttance from an axally [] and a non-axally [2] excted chral sculptured zrcona thn flm. In ths paper we report on the nfluence of dspersve and non-dspersve delectrc functons on the crcular reflectance and transmttance spectra n both axal and non-axal propagaton states by usng the expermental data of refractve ndex of zrcona thn flm. 2. Theory Consder that a regon z d n space s occuped by a chral sculptured thn flm CSTF and that ths flm s beng excted by a plane wave whch propagates wth an 2

3 angle θ relatve to z axs and angle ψ relatve to x axs n xy plane. The phasors of dent reflected and transmtted electrc felds are gven as [3]: E E E ref tr s p s + p + + r = [ al a ] e 2 2 s p s + p r = [ rl + r ] e 2 2 s p s + p + + K r = [ t L t ] e 2 2 K. r K. r. r d u z z z z d The magnetc feld s phasor n any regon s gven as: H r = ωµ E r where a a L L r r and t t are the ampltudes of dent plane wave and L reflected and transmtted waves wth left- or rght-handed polarzatons. We also have; r = xu x + yu y + zu z K = K snθ cosψ u x + snθ snψ u y + cosθ u z 2 where K = ω µ ε = 2π / λ s the free space wave number λ s the free space 2 wavelength and ε = Fm and 7 µ = 4π Hm are the permttvty and permeablty of free space vacuum respectvely. The unt vectors for lnear polarzaton parallel and normal to the dent plane s and p respectvely are defned as: s = snψ u p = mcosθ x + cosψ cosψ u u y + cosθ snψ + snθ ± x y z u u 3 and u x y z are the unt vectors n Cartesan coordnates system. The reflectance and transmttance ampltudes can be obtaned usng the contnuty of the tangental components of electrcal and magnetc felds at two nterfaces z = and z = d and solvng the algebrac matrx equaton [3]: 3

4 4 [ ] [ ] [ ] Ω Ω = + L L L L L L r r r r a a a a K d M d B K t t t t ψ θ ψ κ ψ θ dfferent terms and parameters of ths equaton are gven n detal by Venugopal and Lakhtaka see equatons n reference [3]. In order to obtan ] [ d M ψ Ω κ the pecewse homogenety approxmaton method [4] s used. In ths method the CSTF s dvded nto N a bg enough number very thn layers wth a thckness of N d h / = 5 nm wll suffce. Once the transmttance and the reflectance ampltudes are obtaned from Eq. 4 then we can obtan the reflectance and transmttance coeffcents as: 5 L a t t a r r = = = The transmttance and reflectance are obtaned from: L t T r = = = 3. Numercal results and dscusson We consder that a rght-handed zrcona sculptured thn flm wth a thckness d n ts bulk state has occuped the free space. The relatve permttvty scalars c b a ε n ths sculptured thn flm were obtaned usng the Bruggeman homogenzaton formalsm [56]. In ths formalsm the flm s consdered as a two phase composte vacuum phase and the luson phase. These quanttes are dependent on dfferent parameters namely columnar form factor fracton of vacuum phase vod fracton the

5 wavelength of free space and the refractve ndex ω k ω n s + of the flm s materal luson. Each column n the STF was assumed to consst of a strng of small and dentcal ellpsods and are electrcally small.e. small n a sense that ther electrcal nteracton can be gnored. Therefore [2]: s s v v λ f γ γ γ γ a b 7 ε = ε ε σ c σ σ s o v τ b τ b = where f s the fracton of vod phase ε ω ω k ω 2 v s n s + = s the relatve delectrc permttvty sv γ τ s one half of the long axs of the luson and vod s v ellpsods and γ s one half of the small axs of the luson and vod ellpsods. b s s v v In all calculatons the followng parameters were fxed; γ b = 2 γ τ = 2 γ b = γ τ = o f =.6 χ = 3 Ω = 62nm d = 4Ω and a range of wavelengths v 25nm 85nm λ was consdered where the real refractve ndex of zrcona n ts bulk state Fg. vares from to for the lowest wavelength to hghest wavelength respectvely []. The man parameters chosen n ths work s namely γ τ f v Ω d are very smlar to those reported by Sherwn et al. [6] for ttanum oxde. The dfference between our other parameters and those of Sherwn et al [6] may be explaned on the bass that n our work we have not reported any expermental results for zrcona whle Sherwn et al obtaned expermental data and ftted that to ther theoretcal work. Ths can be admttedly consdered as one of the weaknesses of our work whch s beng consdered for future studes. It should be noted that the magnary part of the refractve ndex for zrcona n ths range of the wavelengths s zero Fg. hence dsspaton can be gnored. In addton t s worthwhle to clarfy that the bulk data for zrcona presented n reference [] s 5

6 remented n nms. In order to carry out our calculatons for each of these wavelengths t was necessary to terate the Bruggeman equaton 2 tmes. In each plot of Fgs. 2 4 and 6 four spectra curves are depcted. In curve the dsperson of delectrc functon s luded n the Bruggman homogenzaton formalsm.e. homogenzaton s mplemented for each wavelength. In curves and v the homogenzaton s performed for Br λ < λ ds Br λ = λ ds and λ > λ s the Bragg wavelength when dsperson of delectrc functon s Br λ ds Br ds taken nto account respectvely. The Bragg wavelength n Fgs.2 4 and 6 s 48 nm 42 nm and 4 nm respectvely whch was obtaned usng Bruggeman formalsm ludng dsperson functon and the data s presented as curve n each fgure. The lower and hgher wavelengths n each case are gven n the fgure captons. Therefore n the latter the permttvty scalars reman the same for other wavelengths. In each plot of Fgs.3 5 and 7 three spectra are depcted curves - - and -v. These spectra show the dfference between the values obtaned n Fgs.2 4 and 6 for dspersed and non-dspersed states of reflectance and transmttance. These dfferences are calculated usng: p q = p q λ p q λ p q = L ; = curve = curves. v 8 where λ and λ present the homogenzaton over the whole wavelength p q p q regon and homogenzaton at a certan wavelength respectvely. In Fg. 2 the crcular reflectance and transmttance spectra and n Fg. 3 the dfferences of the values obtaned n Fg. 2 for crcular reflectance and transmttance between dspersve and non-dspersve states for a rght-handed zrcona CSTF as a functon of wavelength λ for θ = ψ = n axal propagaton state z axs are gven. 6

7 It s well known that for an axally excted CSTF L = L and L TL T = ;. e. the corss-polarzed reflected and transmtted ntenstes do not show any dependence on the crcular polarzaton state of the dent plane wave. Ths s a consequence of the relatve permttvty dyadc [2] of the CSTF beng symmetrc [7]. The results presented n Fgs.2 and 3 can be nterpreted as follows: a reflectance: n the LL plot se the structural handedness of the thn flm s not the same as the polarzaton of the dence plane wave Bragg peaks are smaller and there s no consderable dfference between dspersve curve and nondspersve states curves and v. It can be observed that LL s neglgble. 2 n the L plot t can be seen that there exst a relatvely a consderable dfference between dspersve and non-dspersve states L and ths dfference reases by movng further away from the homogenzaton wavelength.e. the wavelength at whch the homogenzaton s beng performed. The homogenzaton wavelength for curves and v were 43 nm 48 nm and 53 nm respectvely. 3 n the plot owng to the same structural handedness n the structural drecton n the thn flm and the polarzaton of the dence plane wave crcular Bragg peaks occur vvdly n the Bragg regon. The dfference between the spectra at wavelengths near to homogenzaton wavelengths s small and becomes zero at homogenzaton wavelength. In order to clearly observe how the Bragg regmes are affected through theses processes the and plots n the wavelength regon of 45 to 55 nm are gven n Fg. 4a-b. In Fg. 4b t can be 7

8 observed that the Bragg regme s most affected at lower wavelengths than that of Bragg. b transmttance: a rght-handed chral sculptured thn flm transmts the LCP plane wave almost completely Fg.2: T LL plot. It can be observed that there exsts a consderable dfference between dspersed and non-dspersed states. Ths dfference s more pronounced at wavelengths further away from the homogenzaton wavelength Fg.3: TLL plot. 2 n T L plot the transmttance spectrum n the Bragg regon stands at hgher values than those outsde Bragg regon. No dfference can be observed between dspersed and non-dspersed spectra Fg.3: TL plot. 3 n T plot n the Bragg regon the transmtted spectrum s mnmzed. At wavelengths further away from the homogenzaton wavelength the dfference between the dspersed and the non-dspersed states become more pronounced Fg.3: T plot. In Fg. 5 the crcular remttances and n Fg. 6 the dfferences of the values obtaned n Fg. 5 for crcular reflectances and transmttances between the dspersve and the nondspersve states for a rght-handed zrcona CSTF as a functon of wavelength λ for o o θ = 45 = ψ n the non-axal propagaton state xy plane are gven. The results gven n Fg. 5 and Fg. 6 may be descrbed as follows: a reflectance n LL plot the crcular Bragg phenomenon character s not obvous but there s a consderable dfference between dspersed and non-dspersed states and at wavelengths further away from the homogenzaton 8

9 wavelength ths dfference becomes more pronounced Fg. 6: LL plot. The homogenzaton wavelength for curves and v were chosen as 37 nm 42nm and 47 nm respectvely. 2 In L plot the frst and second Bragg peaks are clearly observed whle the hgher order peaks occur at shorter wavelengths wth much less strength hence they cannot clearly be observed. There s a consderable dfference between dspersed and non-dspersed states whch s more pronounced at longer wavelengths.e. where the frst Bragg peak appears Fg. 6: L plot. 3 The dscusson and nterpretaton of the results gven for L Fg. 5 and In L Fg. 6 are applcable to L Fg. 5 and L Fg. 6 wth a lttle dfference between spectral values. Ths shows that the CSTF dscrmnates between rght crcularly and left crcularly polarzed plane waves. plot the occurrence of crcular Bragg phenomenon s more dstngushable. Also the dfference between dspersed and non-dspersed states curves and v n the Bragg regon s more pronounced Fg. 6: plot. The reason for small dfference between the non-dspersed state curve and the dspersed state curve s due to the fact that n curve the homogenzaton has been performed at only 42 nm wavelength whch s the same wavelength for whch the reflectance spectrum n dspersed state s maxmum. In order to observe The nfluence of the dent angle θ on the Bragg regmes the and plots n the wavelength regon of 4 to 5 nm are gven n Fg. 7a-b. It can be observed n Fg. 7b that smlar to Fg. 4b the Bragg regme s most affected at 9

10 lower wavelengths than that of Bragg. However comparson of Fg. 7b wth Fg. 4b shows that the nfluence of the change n the dent angle from zero degree to 45 degrees has affected the Bragg regme by a factor of ten. b transmttance: T LL spectra are dsordered and the effect of dsperson of delectrc functon can be clearly dstngushed n the regons far away from the homogenzaton wavelength Fg. 6: TLL plot. 2 There exst a trough n the T L spectra whch corresponds to the second Bragg peak. The nfluence of the dsperson of delectrc functon at longer wavelengths s more pronounced Fg. 6: TL plot. 3 The dscusson and nterpretaton of the results gven above for T L and T L are applcable to T L and TL wth a lttle dfference between spectral values. Ths shows that the CSTF dscrmnates between rght crcularly and left crcularly polarzed plane waves. 4 Two troughs can be observed n T spectra. The dfference between the dspersed and non-dspersed delectrc functons become more vvd at wavelengths further away from the homogenzaton wavelength Fg. 6: T plot. In Fg. 8 the crcular remttances and n Fg. 9 the dfferences of the values obtaned n Fg. 8 for crcular reflectance and transmttance between dspersve and nondspersve states for a rght-handed zrcona CSTF as a functon of wavelength λ for o o θ = 45 = 9 ψ n non-axal propagaton state yz plane are gven. The comparson of Fgs. 8 and 9 wth Fgs. 5 and 6 respectvely shows that n fact the obtaned spectra are almost smlar and the only dfference s that n Fgs. 8 and 9 the

11 homogenzaton for curves and v s performed at 36 nm 4 nm and 46 nm wavelengths. The fundamental dfference between dspersve and non-dspersve states s n Fg. 9 curve - whch s clearly dstngushable at homogenzaton wavelength. Fg. shows the plots of and n the wavelength regon of 4 to 5 nm. Fg. b agan shows the nfluence of the shorter wavelengths on the Bragg regme whle hgher wavelengths have lesser effect In summary n ths work by usng the Bruggeman homogenzaton formalsm and the pecewse homogenety approxmaton method we have been able to show the nfluence of the dsperson of delectrc functon n reflectance and transmttance spectra of crcularly polarzed plane waves from a rght-handed zrcona CSTF for both axal and non-axal propagaton states. The nfluence of dsperson effect on axal exctaton of cross-polarzed reflectances and co-polarzed transmttances becomes more detectable at wavelengths further away from the homogenzaton wavelength. For non-axal exctaton of crcular transmttances smlar results to those of axal exctaton are obtaned. There exst fundamental dfferences for cross-polarzed reflectances where wavelength the frst Bragg peak occurs. The dsperson effect of the delectrc functon n becomes more sgnfcant at wavelengths near the homogenzaton wavelength. 4. Conclusons The nfluence of the dsperson of delectrc functon n the remttances spectra of crcularly polarzed plane waves from a rght-handed zrcona CSTF s reported usng the pecewse homogenety approxmaton method and the Bruggeman homogenzaton formalsm at each gven frequency for both axal and non-axal

12 propagaton states. Ths was carred out by consderng the refractve ndex of zrcona at each gven frequency ndvdually n the frequency range of 25 to 85 nm n the homogenzaton formalsm. Therefore n ths way dsperson of the delectrc functon was ntroduced nto our calculatons. Ths method drectly takes advantage from the expermental relatve delectrc constant of thn flm and avods the use of smple dsperson model known as sngle-resonance Lorentzan model because t s beleved that not all thn flm systems may obey the sngle-resonance Lorentzan model but may be composed of a double-resonance or multple-resonance systems. Acknowledgements Ths work was carred out wth the support of the Unversty of Tehran and the Iran Natonal Scence Foundaton INSF. eferences [] F. Babae and H. Savalon Opt. Commun [2] F. Babae and H. Savalon Opt. Commun [3] A. Lakhtaka Mcrow. Opt. Technol. Lett [4] Fe. Wang and A. Lakhtaka Opt. Commun [5] Joseph B. Geddes III and A. Lakhtaka Opt. Commun [6] Fe. Wang and A. Lakhtaka Opt. Commun [7] S. Shen and K. E. Oughstun J. Opt. Soc. Am. B [8] P. E. Tannewald and M. H. Seavey J. Phy. ev [9] A. ArreanzV. Perez-Deste and C. Palaco Phy. ev. B [] E. D. Palk Handbook of Optcal Constants of SoldsAcadamc press Newyork985. [] F. Babae and H. Savalon J. Mod. Optcs n press [2] F. Babae and H. Savalon submtted to J. Mod. Optcs [3] V. C. Venugopal and A. Lakhtaka Proc.. Lond. A [4] A.Lakhtaka and.messer n Sculptured thn flms: Nanoengneered morphology and optcs SPIE Press Bellngham WA USA 25Chap9 [5] J.A. Sherwn and A. Lakhtaka Math and Compu. Model ; J.A. Sherwn and A. Lakhtaka Math and Compu. Model ; [6] J.A. Sherwn A. Lakhtaka and I.J. Hodgknson Opt. Commun [7] V.C. Venugopal A. Lakhtaka Opt. Commun ; V.C. Venugopal A. Lakhtaka Opt. Commun

13 Fgure captons Fgure. The refractve ndex of pure bulk zrcona showng that the magnary part s zero n the wavelength regon shown. Fgure 2. eflectance and transmttance spectra from a rght-handed zrcona CSTF as o a functon of wavelength λ for θ = ψ = n axal propagaton z axs. a reflectance; b transmttance.the calculatons performed for curves to v accordng to the followng condtons: n s 25nm85nm [ ] n s 43nm = n s 48nm = v n s 53nm = Fgure 3. The dfferences obtaned between dspersve and non-dspersve states of the results presented n Fg. for reflectances and transmttances. Fgure 4. Plots of a and b as n fgures 2 and 3 but drawn n the wavelength range of 45 to 55 nm. Fgure 5. eflectance and transmttance spectra from a rght-handed zrcona CSTF as o o a functon of wavelength λ for θ = 45 ψ = n non-axal propagaton xz plane. a reflectance; b transmttance. The calculatons performed for curves to v accordng to the followng condtons: n s 25nm85nm [ ] n s 37nm = n s 42nm = v n s 47nm = Fgure 6. The dfferences obtaned between dspersve and non-dspersve states of the results presented n Fg. 3 for reflectances and transmttances. Fgure 7. Plots of a and b as n fgures 5 and 6 but drawn n the wavelength range of 4 to 5 nm. Fgure 8. eflectance and transmttance spectra from a rght-handed zrcona CSTF as o o a functon of wavelength λ for θ = 45 ψ = 9 n non-axal propagaton yz plane. a reflectance; b transmttance. The calculatons performed for curves to v accordng to the followng condtons: n s 25nm85nm [ ] n s 36nm = n s 4nm = v n s 46nm = Fgure 9. The dfferences obtaned between dspersve and non-dspersve states of the results presented n Fg. 5 for reflectances and transmttances Fgure. Plots of a and b as n fgures 8 and 9 but drawn n the wavelength range of 4 to 5 nm. 3

14 n k λ nm Fg.; F. Babae and H. Savalon 4

15 a eflectance b Transmttance E-3 LL T LL L T L λ v T λ v Fg.2; F. Babae and H. Savalon 5

16 a Dfferences n reflectances b Dfferences n transmttances E-3 LL T LL L T L λ - -v - T λ - - -v Fg.3; F. Babae and H. Savalon 6

17 .9 a v λ..5 b λ - - -v Fg. 4.; F. Babae and H. Savalon 7

18 a eflectance b Transmttance LL T LL L T L L T L λ v T λ v Fg. 5; F. Babae and H. Savalon 8

19 a Dfferences n reflectances b Dfferences n transmttances LL T LL L T L L T L λ - - -v T λ - - -v Fg. 6; F. Babae and H. Savalon 9

20 a v λ b - - -v λ Fg.7.; F. Babae and H. Savalon 2

21 a eflectance b Transmttance LL T LL L T L L T L λ v T λ v Fg. 8; F. Babae and H. Savalon 2

22 a Dfferences n reflectances b Dfferences n transmttances LL T LL L T L L T L λ - - -v T λ - - -v Fg. 9; F. Babae and H. Savalon 22

23 .9.8 a.6.5 b v v λ λ Fg..; F. Babae and H. Savalon 23