Optik kuplörlerin tasarımı için bir yöntem

Abstract

Bu çalışmada kuple optik dalga kılavuzlarının pro- pagasyon sabitlerinin belirlenmesi problemi incelenmekte dir. Kuplörün tasarımı için gerekli olan kuplaj katsayı sı ve kuplaj mesafesinin hesapl anabil mesi için propagas- yon katsayılarının yüksek doğrul ukl a el de edilmesi gere kir. Kuple optik dalga kılavuzlarında tam çözümlerin el de edilmesindeki güçlükler nedeniyle, bu problemin çözü münde genellikle yaklaşık yöntemlerden yararlanmak zorunlu olur. Çalışmada, bu amaçla varyasyonel yöntem ve iteratif yöntem adı verilecek olan iki farklı yol izlenmiştir. İlk yöntemde, küple sistemin propagasyon sabiti için elde edilen varyasyonel bir ifadeden hareketle tam çözümleri daha kolay elde edilebilen test problemleri yardımı ile propagasyon sabitlerinin yaklaşık ifadelerinin bulunması amaçlanmıştır. İkinci yöntemde ise, küple dalga kılavuzu sisteminin belirli bir modu ele alınarak, bu moda ili şk in özdeğer ile aynı özdeğere sahip fiktif bir dalga kılavuzu nun belirlenmesi hedeflenmektedir. Her iki yönteme iliş kin uygulama örneklerinde elde edilen yaklaşıklıkların iyilik derecelerini belirleyebilmek amacıyla, tam çözümü kolayca bulunabilen optik dalga kılavuzlarından oluşan kuplörler göz önüne alınmıştır.

Coupled optical waveguides have great importance in theoretical and experimental works in various areas of guided wave optics in optical communication systems and in integrated optical systems. To generate solutions one ge nerally resorts the coupled-mode theory which is approxi mate and the approximation is not always sufficiently close to the exact solution and hence cannot be applied directly to the design of optical couplers. The most important design parameter is the propagation constant which determines coupling lengths and coupling coefficients. This thesis deals with the problem determi ning accurate approximations to the propagation constants of coupled optical waveguides. in two different ways. To achieve this goal, two different approaches are investigated to be referred to as the variational method and the iterative method, respectively. The variational method is based on a recent formulation of coupled-mode theory developed by Haus et.al. [4]. Classical coupled- mode theory is an approximation wherein the transverse com ponents of eigenfuncti ons of the coupled system are repre sented by a superposition of eigenfunctions of the isolated waveguides. As discussed in the second section of this thesis to reduce the error caused by this approximation Haus et.al. use a variational expression for the eigen values to obtain coupled mode equations which contain also the axial components of eigenfunctions of the isolated waveguides. However, even in this improved representation, approximation is not sufficiently close to the exact solu tion. In particular, the error incurred in the calculation of the odd mode eigenvalues can be as high as 10"1. The eigenfunctions of isolated waveguides for odd symmetric modes which are substituted to the variational expression can be tought as the solutions of properly chosen test problems. In this thesis, we investigated the possibility of improving the accuracy of this approach by chosing the test problems, which are different from that formed by the isolated waveguides. For this purpose we have constructed two test problems which are discussed in the third section. (v) The formulation of the second ve method is given in the second se based on determining a fictive equi guide a parameter of which is pertu to render eigenvalues which are ide coupled system. The iterative meth problem of determining the eigenval films in the third section. Since a fictive equivalent isolated waveg «above mentioned special case, the i thought as a fast and simple altern lution of the eigenvalue equation o wavegui des. approach, t eti on. This valent isol a rbed in such ntical with od is applie ues of coup! the determin uide is exac terative met ative to the f the couple he iterati- method is ted wave - a way as that of the d to the :> e d optical at i on of t in the hod can be direct so- d system of Figure 1. Coupled Optical Waveguide In the second section of this thesis, lossless coup led optical film waveguide is considered as a canonical problem for which exact solutions are easy to generate. Assuming the field is independent of the x coordinate, and the propagation direction is along z, die.îectric constant function e(y) is written as, e(y) - eon2(y; (1) The electric and magnetic fields obey Maxwell's equa tions. The operator V can be seperated into longitudinal and transverse components, (vi) az + v, (2) We consider a wave propagating in the pozitive z direction and write exp(-jgz) with 3-real. Maxwell's equations can then be written as VtxE + jwy0H =j$uzxE VtxH - jwe E =j3uzxH (3) (4) One can solve for B by cross multiplying (3) by H and (4) by if*. subtracting the resulting equations and integrating over the waveguide cross section. B = { 1_ I ((Vtxî+ jwji0"M* -(Vtxti-jwet)"E*)ds } ll J,(ExH *¦ E*xH).J2ds} (5) where we use the symbol j ds for the double integration (ds=dxdy) over the waveguide cross section. This is a va riational expression for B. Indeed, if one denotes by E0 and Ho the exact solution with the propagation constant Bo. and if one inserts into (5) a perturbed field E= E"+3E H =Ho + 3H 2 2 the value of B changes only to order d£ and 8H. The proof of this statement is given in the second section, The coupled mode equations are obtained from (5) by substituting the trial solutions N f-I a.e. 1-1 ' ' N HS=laihi (6) (7) (vii) where e. and ti. are normalized solutions for the electric and magnetic fields, respectively of the modes of a N-guide system, obeying the equations Vtxei + jwy0h. - jB^u xe. V*i " J'w^-iei - jHiuzxh. (8) (9) Here e^ defines the refractive index distribution of the ith waveguide. The choice of e- fixes Î and ft through (6) and (7) except for expansion coefficients a.. When (6) and (7) are introduced into (5) one obtains ;. «*HU'.1 -T"pu aj (10) where the Einstein convention of summation over repeated indices is used and Pij " ı r if*. z* (e. x h? + e~ x h.) u2ds (11) and Hid 1 P..g. + - 10 J 4 w (e - e1)e1e1 ds (12) The matrix P.. is Hermitian because a^p-j-ia-j ^s time average power and"Ms thus real. J J For a system of waveguides with identical real propa gation constants it is obvious that H.. is also Hermitian But this property is preserved even when £-;=£,. and/or 3rar J H..= H* (13) The optimum value of g under an assumed trial solution is obtained by extremizing (10). Thus one must differen tiate (10) with respect to the magnitidus [a-[ and phases (J» j of the complex amplitudes a.. Differentiation with res pect to . and | si -| J is equivalent to di f ferentiotion with respect to the complex amplitude a* or a-. (viii) Differentiation with respect to a. gives BPUaJ-H1j»J (14) Differentiation with respect to a. gives J *PUaî-H1J«î (15) The two equations give the same determinantal equation fo r 3 if, and only if, P.. and H-. are Hermitian det(3P..-H.. ) = 0 (16) In the variational method, focusing our attention to optical film couplers we have investigated the possi bility of obtaining accurate solutions via (16), by uti lizing test functions in (6) and (7) which are defined by fictituous isolated films with suitably modified parameters The iterative method presented in this thesis can best be explained on hand of transverse resonance equivalent transmission line concept. Thus assuming that such an equivalance can be cunstructed to the problem under inves tigation of the determination of the eigenvalues reduces to solving ?(3) + î(3)- 0 (17) for 3, where 1 and Î stand for the impedances of the modal transmission lines at an arbitrary point when looking in one or the other direction. 3 should be determined via (17) upon replacing L and 1 by X and Jt respectively. We assume that for a given coup! ed~waveguide system one can costruct a test problem with the following two properties: 1. ?(3)==^(3) i.e. when looking in one of the di rections modal impedances have identical functional forms in terms of the physical and electrical properties of the two problems. 2. The equation ?($)¦= ?(3) can be solved for a para meter, say A(3) of the test problem. Under above assumptions following iterative algorithm can be constructed: (ix) a. Determine A(3) and hence the test problem using an initial guess for 3. b. Determine the successive approximation to 3» through the solution of the test problem. c. Feed 3 obtained in (b) to (a) and repeat the iteration. Three examples for the appl ication of this method to optical film couplers are given in Chapter 3. Equality signs in both 1. and 2. apply for the structures consi dered in this thesis. The presented results are therefore exact but of limited significance. The applicability of the method to more general problems wherein 1. and 2. can not be satisfied exactly is yet to be investigated.