A semi-analytic technique to determine the propagation constant of periodically segmented Ti:LiNbO 3 waveguide

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1 Avalable onlne at Pelaga Research Lbrary Advances n Appled Scence Research, 011, (1): ISSN: CODEN (USA): AASRFC A sem-analytc technque to determne the propagaton constant of perodcally segmented T:LNbO wavegude Pranabendu Ganguly Advanced Technology Development Centre, Indan Insttute of Technology, Kharagpur, Inda ABSTRACT A sem-analytcal technque to determne the propagaton constant of perodcally segmented T:LNbO wavegude (PSW) s descrbed. The -D refractve ndex profles of the wavegude segments are computed from ts fabrcaton parameters, such as, T-layer thckness and wdth, and the dffuson parameters. WKB method s used to transform -D refractve ndex profle to 1- D lateral effectve ndex profle, whch s then converted to equvalent refractve ndex profle of the PSW. Fnally transfer matrx method s appled to compute the propagaton constant of the wavegude. Key words: T:LNbO, Perodcally segmented wavegude, Effectve-ndex, Equvalent refractve ndex, Matrx method. INTRODUCTION There has been an ncreasng amount of nterest n the applcaton of perodcally segmented wavegudes (PSW s) n ntegrated optcs. To date these devces have been reported n a number of dfferent materal systems ncludng proton-exchanged LNbO [1], KTP [], InP [], annealed proton exchanged LNbO [4-6] and T:LNbO [7-9]. Applcaton of PSWs ranges from nonlnear devces, whch uses quas phase matched second harmonc generaton [], to lnear devces lke a -D mode taper [10], asymmetrc drectonal coupler flter [11], and wavelength demultplexer [9]. In a PSW the ncrease n refractve ndex ( n) s modulated perodcally durng fabrcaton, as shown n Fg.1. Pelaga Research Lbrary 16

2 Pranabendu Ganguly Adv. Appl. Sc. Res., 011, (1): Fgure 1 Perodcally segmented wavegude (PSW). As a consequence of the segmentaton, the loss n the gude s ncreased and the effectve refractve ndex s reduced when compared to a contnuous wavegude. A PSW s characterzed by ts perod, Λ, and duty-cycle, η (the rato of the length of a segment and the perod of the gude). It has been demonstrated that a PSW can be represented by an equvalent contnuous wavegude wth same depth and wdth, n whch the average ndex dfference ( n eq ) s taken to be the weghted average of the ndex along the propagaton drecton. Ths s represented by equaton (1), [1] n eq = η n (1) By choosng the duty-cycle, the average ndex can be spatally modfed along the wavegude. The refractve ndex change determnes the mode sze, propagaton constant and cut-off wavelength of the PSW. Perodcally segmented wavegudes formed by T-ndffuson n a LNbO substrate wll have a graded concentraton varaton along depth and lateral drectons. In ths work, the propagaton constants of perodcally segmented T:LNbO wavegudes have been computed by applyng the effectve-ndex-based matrx method (EIMM) along wth equvalent wavegude concept. In the frst step, the depth and lateral refractve ndex profles of each T:LNbO segment has been computed drectly from ts fabrcaton parameters, such as, T-layer thckness and wdth, and the dffuson parameters. Effectve ndex method has been appled to transfer the -dmentsonal ndex dstrbuton to 1-dmensonal lateral effectve-ndex dstrbuton. In the next step, the average refractve ndex has been computed to transfer the PSW nto an equvalent contnuous wavegude. Fnally, the transfer matrx method s used to compute the propagaton constant of the equvalent wavegude. Theoretcal Approach Determnaton of effectve refractve ndex of T:LNbO PSW In the case of T:LNbO wavegudes the T-concentraton profle can be represented by [1] 1 τ z τ z C ( x, z) = C o erf erf 4 d z d z Pelaga Research Lbrary 17

3 Pranabendu Ganguly Adv. Appl. Sc. Res., 011, (1): W x W x erf erf () d x d x where C o s the sold solublty of T nto LNbO, W and τ are the deposted T strp wdth and thckness, and d x and d z are the dffuson lengths along x and z axes of the crystal The refractve ndces of ordnary and extraordnary ray, n o and n e, of the congruently grown LNbO crystal, at room temperature (5 o C), can be obtaned usng the modfed Sellmeer equatons [14] n = λ λ o () ( ) λ n e = λ (4) ( ) where λ s the wavelength n µm. The refractve ndex change nduced by T-ndffuson for both ordnary and extraordnary rays are related to the ttanum concentraton for LNbO as [1] α o, n x z, λ = A ( λ) C ( x, z) (5) o, e ( ) [ ] e, o, e where A o,e are dependent on λ, and α o,e are 0.5 and 0.85 for ordnary and extraordnary rays. Detals of the λ dependence of A o,e are gven n ref.[1], whch are vald wthn the wavelength range 0.6 λ (µm) 1.6. So by usng equatons (-5) one can determne the -dmensonal refractve ndex dstrbuton of the segment of a T:LNbO PSW drectly from ts fabrcaton parameters. Now a wavegude mode exsts only f the total transverse phase shft (along Z) for one round-trp across the gude equals an ntegral multple of п,.e., [15] Zb 1/ [ n ( z, x) n ( x) ] dz φ φ = mπ 4π eff t b λ 0 m = 0,1,,K (6) Ths refers to the wave confnement over the YZ plane at a partcular lateral poston (x). The frst term s the total phase change of the wave as t travels between the crystal surface z = 0 and the return pont z = Z b n the bulk and back. ф t and ф b are Goos-Hanchen phase shfts correspondng to total nternal reflecton at z = 0 and z = Z b, respectvely. For the LNbO -ar nterface, ф t s about -0.9п and -0.98п for TE and TM modes, respectvely, and ф b approaches (-п/) for both polarzatons [15]. Equaton (6) s known as the WKB quantzaton condton. Ths leads to a set of dscrete angles of propagaton correspondng to dfferent guded modes. So by solvng equaton (6) numercally for the fundamental mode (m = 0) one can determne the effectve refractve ndex, n eff (x), of the sngle-mode T:LNbO wavegude. For the computaton of refractve ndex of equvalent wavegude for a PSD wth duty-cycle η one has to use equaton (1),.e., n eq ( x) = η n ( x) (7) eff In the next step, ths n eg (x) s approxmated by a starcase-type step-ndex profle and matrx method s appled to that layered structure. Pelaga Research Lbrary 18

4 Pranabendu Ganguly Adv. Appl. Sc. Res., 011, (1): Determnaton of propagaton constant The propagaton constants of the T:LNbO PSW s are here determned by matrx method. If the medum s consdered to be made of a number of layers (each of thckness d and refractve ndex n ), then for an ncdent plane wave n the frst layer, the electrc feld assocated wth each layer can be represented as [15] E = u E exp( j ) exp j ωt ( K cosθ ) x β y where E and u E [ { }] j )exp[ j{ ωt ( K cosθ ) x β y} ] exp( (8) 1 = = 0 ; = K d cosθ ; 4 = K 4 ( d d ) cosθ 4 ; K ( d d d )cos π = L 1 θ ; K = K o n = n ; λ β = K snθ. (9) E are the magntude of electrc feld vectors assocated wth the downward and upward-propagatng waves, respectvely, and u and u are ther correspondng unt vectors, and β s the propagaton constant along the layers and s the same for all the layers, whch s consstent wth the law of refracton. Applyng the boundary condtons at the nterfaces, one obtans E 1 E E N = = = S 1 L S1 S LS N 1 (10) E1 E E N where 1 exp( jδ ) r exp( jδ ) S 1 = (11) t r exp( jδ ) exp( jδ ) π δ = n d cosθ λ, =,,4,,N. o S -1 s the transmsson matrx from the (-1)-th and -th layer; r and t represent the ampltude reflecton and transmsson coeffcents, respectvely, from the (-1)-th layer to the -th layer. Usng equatons (8-11) one can determne the feld ampltude n any layer n terms of the ncdent feld ampltude E. 1 The above method when appled to equvalent refractve ndex profle of equaton (7) can be used to determne the propagaton constant of T:LNbO perodcally segmented wavegudes. Prsm couplng approach has been appled for the purpose. Intally a hgher refractve ndex layer s consdered as layer 1 and exctaton effcency of the gudng layer s computed by matrx method for dfferent ncdent angles. Out of the dfferent ncdent angles only a sngle wll excte the guded mode n the sngle-mode wavegude refractve ndex structure. Exctaton effcency versus ncdent angle or propagaton constant characterstcs wll show only one peak for the guded mode, from whch propagaton constant of the PSW can be determned. The detals of the matrx method are gven n ref. [16]. Pelaga Research Lbrary 19

5 Pranabendu Ganguly Adv. Appl. Sc. Res., 011, (1): Computatonal Procedure and Results Computatonal Procedure The computer program wrtten for calculatng propagaton constant of T:LNbO PSW requres the followng parameters as nputs: 1. T flm parameters: thckness, strp wdth, gap between the segments, and length of segments,. T dffuson parameters: dffuson temperature, tme, and ambent,. Beam parameters: wavelength and polarzaton (TE / TM), 4. Computatonal parameters: samplng area (x range, z range) and the layer thcknesses of the dscretsed n eq (x) profle. The frst three sets of parameters are taken as the fabrcaton parameters of T:LNbO PSW, whle the x range and z range are taken as -15 to 15 µm and 0 to 15 µm, respectvely, for most computatons. A unform layer thckness of 0.01 µm has been used. However, n order to reduce the computaton tme wthout loss of accuracy, nonunform parttonng of n eq (x) nvolvng a fewer number of layers may also be consdered. The computatonal steps to determne the propagaton constants of T:LNbO PSWs are as follows: 1. Computaton of two-dmensonal refractve ndex profles, n(x, z), of T:LNbO wavegude from ts fabrcaton parameters.. Computaton of one-dmensonal lateral effectve ndex profle, n eff (x), from the twodmensonal refractve ndex dstrbuton by WKB method.. Determnaton of equvalent refractve ndex profle, n eq (x), for the PSW from n eff (x). 4. Applcaton of matrx method onto n eq (x) to determne the propagaton constant of T:LNbO PSW. Followng the matrx method outlned n prevous secton, the exctaton effcency s computed for varous values of ncdence angle θ 1 from frst hgh refractve ndex medum (added) for θ c θ 1 90 o to fnd the values of θ 1 and hence β [usng equaton (9)] for whch exctaton effcency shows sharp resonance peaks. The gap between the added medum and the boundary of the wavegude s ncreased untl the lmtng values of β are obtaned. The flow chart of the entre computatonal procedure s gven n Fg.. Pelaga Research Lbrary 140

6 Pranabendu Ganguly Adv. Appl. Sc. Res., 011, (1): User Screen START Computaton of r.. of the wavegude from ts fabrcaton parameters Converson of D r.. profle to 1D effectve ndex profle by WKB method Computaton of n eq (x) from the effectve ndex profle INPUTS x, n 1, d Determnaton of θ C for the structure θ = θ C Add ncrement to θ Generaton of exctaton effcency versus β data No Is θ 90 o? Yes Determnaton of exctaton effcency versus β values for the resonance peak OUTPUT β value of PSW Fgure Flow-chart for computaton of the propagaton constant of T:LNbO PSW. Pelaga Research Lbrary 141

7 Pranabendu Ganguly Adv. Appl. Sc. Res., 011, (1): RESULTS AND DISCUSSION The computed propagaton constant versus gap between the segments of T:LNbO PSW s shown n Fg. for TM mode at 1.1 µm transmttng wavelength. The Propagaton Constant (µm -1 ) Length of each segment = 10 µm T-thckness = µm T-wdth = 6. µm Dffuson temperatute = 1050 o C Dffuson tme = 6.0 hours Wavelength = 1.1 µm, TM mode Gap between the segments (µm) Fgure Computed propagaton constant versus gap between the segments of a T:LNbO PSW. Pelaga Research Lbrary 14

8 Pranabendu Ganguly Adv. Appl. Sc. Res., 011, (1): Segment length = 10.0 µm Gap between the segments = 9.5 µm Exctaton Effcency Propagaton Constant (µm -1 ) Fgure 4 Typcal resonance peak of exctaton effcency versus propagaton constant plot of T:LNbO PSW. length of the segments s taken as 10 µm. It may be noted from the fgure that as the gap between the segments ncreases the propagaton constant decreases. Ths nature of the characterstc s obvous from the fact that as gap between the segments ncreases, the n eq of PSW decreases and hence propagaton constant of the wavegude decreases. Typcal exctaton effcency versus propagaton constant plot for gap 9.5 µm s shown n Fg.4, whch shows a sharp peak ndcatng the guded mode propagaton constant of the PSW. It may be noted that the present analytcal model s usable for both TE and TM polarzed lghts for dfferent sets of PSW fabrcaton parameters wthn a wavelength range from 0.6 to 1.6 µm. The method nvolves only straghtforward multplcaton of x transfer matrces of the layers. It does not requre the soluton of any transcendental or dfferental equaton. Iteratons are used only once to evaluate the effectve refractve ndex of the wavegude. Ths makes the present model very fast and smple to use wth a PC. The computed propagaton constant wll be an essental nput for desgn consderaton for several PSW components of optcal ntegrated crcuts, such as asymmetrc drectonal couplers and Y-junctons. Pelaga Research Lbrary 14

9 Pranabendu Ganguly Adv. Appl. Sc. Res., 011, (1): CONCLUSION Propagaton constants of T:LNbO perodcally segmented wavegudes (PSW s) are determned from ts fabrcaton parameters by effectve ndex based matrx method (EIMM) along wth equvalent contnuous wavegude model. The -D refractve ndex profles of the wavegude segments are computed from ts fabrcaton parameters. WKB method s used to transform -D refractve ndex profle to 1-D lateral effectve ndex profle, whch s then converted to equvalent refractve ndex profle of the PSW. Fnally transfer matrx method s appled to compute the propagaton constant of the wavegude. Ths sem-analytcal technque s computatonally fast and smple to use wth a PC. REFERENCES [1] E.B. Pun, K.K. Wong, I. Andonovc, P.J.R. Laybourn, and R. De La Rue, Electron. Lett., 18, pp , 198. [] J.D. Berlen, D.B. Laubacher, and J.B. Brown, Appl. Phys. Lett., 56, pp , [] F. Dorgeulle, B. Mersal, S. Francos, G. Herve-Gruyer, and M. FlocheOpt. Lett., 0, pp , [4] M.H. Chou, M.A. Arbone, and M.M. Fejer, Opt. Lett., 1, pp , [5] K. Thyagarajan, C.W. Chen, R.V. Ramaswamy, H.S. Km, and H.C. Cheng, Opt. Lett., 19, pp , [6] D. Nr, S. Ruschn, A. Hardy, and D. Brooks, Electron. Lett., 1, pp , [7] D. Nr, Z. Wessman, S. Ruschn, and A. Hardy, Opt. Lett., 19, pp , [8] J. Webjorn, F. Laurell, and G. Arvdsson, J. Lghtwave Technol., 7, pp , [9] Z. Wessman, D. Nr, S. Ruschn, and A. Hardy, Appl. Phys. Lett., 67, pp.0-04, [10] Z. Wessman and I. Hendel, J. Lghtwave Technol., 1, pp , [11] Z. Wessman, F. Sant-Andre, and A. Kevorkan, Proc. ECIO 97, pp.5-55, (Aprl 1997). [1] D. Ortega, R.M. De La Rue, and J.S. Atchson, J. Lghtwave Technol., 16, pp.84-91, [1] P. Ganguly, D.C. Sen, S. Dutt, J.C. Bswas, and S.K. Lahr, Fber and Integrated Optcs, 15, pp , [14] D.S. Smth and H.D. Rccus, Opt. Commun., 17, pp.-5, [15] P. Ganguly, J.C. Bswas, and S.K. Lahr, Fber and Integrated Optcs, 17, pp , [16] A.K. Ghatak, K. Thyagarajan, and M.R. Shenoy, J. Lghtwave Technol., 5, pp , Pelaga Research Lbrary 144