Dielectric Rectangular Waveguide and Directional Coupler for Integrated Optics By E.A.J.MARCATILI (Manuscript received March 3,1969) We study the transmission properties of a guide consisting of a dielectric rod with rectangular cross section,surrounded by several dielectrics of smaller refractive indices.This guide is suilable for integrated optical circuitry because of its size,single-mode operation,mechanical stability, simplicity,and precise comstruction. After making some simplifying assumptions,we solve Maxwell's eguations in closed form and find,that,because of total internal reflection, the guide supports two types of hybrid modes which are essentially of the TEM kind polarized at right angles.Their attenuations are comparable to that of a plane wave traveling in the material of which the rod is made. If the refractive indexes are chosen properly,the guide can support only the fundamental modes of each family with any aspect ralio of the guide cross section.By adding thin lossy layers,the guide presents higher loss to one of those modes.As an alternative,the guide can be made to support only one of the modes if part of the surrounding dielectrics is made a low im- pedance medium. Finally,we determine the coupling between parallel guiding rods of slightly different sizes and dielectrics;at wavelengths around one micron, S-dB directional couplers,a few hundred microns long,can be achieved with separations of the guides about the same as their widths (a few microns). I.INTRODUCTION Proposals have been made for dielectric waveguides capable of guiding beams in integrated optical circuits very much as waveguides and coaxials are used for microwave circuitry.-Figure 1 shows the basic geometries for these waveguides.The guide is a dielectrio rod of refractive index n immersed in another dielectric of slightly smaller refractive index n(1-A);both are in contact with a third dielectrio which may be air (Fig.1a)or a dielectric of refractive index n(1-A), 2071
Dielectric Rectangular Waveguide and Directional Coupler for Integrated Optics By E. A. J. MARCATILI (Manuscript received March 3,1969 ) We study the transmisaion properties of a guide consisting of a dielectric rod with rectangular cross section, surrounded by several dielectrics of smaller refrcutive indices. This guide is suitable for integrated optical circuitry because of its size, single-mode operation, mechanical stability, simplicity, and precise construction. After making some simplifying assumptions, we salve MaxvieU's equations in closed form and find, that, because of total internal reflection, the guide supports two types of hybrid modes which are essentially of the TEM kind polarized (U right angles. Their attenuations are comparable to that of a plane wave traveling in the material of which the rod is made. If the refractive indexes are chosen properly, the guide can support only the fundamental modes of each family with any aspect ratio of the guide cross section. By adding thin lossy layers, the guide presents higher loss to one of those modes. As an alternative, the guide can be made to support only one of ike modes if part of the surrounding dielectrics is made a low impedance medium. Finally, we determine the coupling between parallel guiding rods of slightly different sizes and dielectrics; at wavelengths around one micron, S-dB directional couplers, a few hundred microns long, can be achieved wüh separations of the guides about the same as their toidths (o few microTu). I. INTHODUCTION Proposals have been made for dielectric waveguides capable of guiding beams in integrated optical circuits very much as waveguides and coaxials are used for microwave circuitry.*"" Figure 1 shows the basic geometries for these waveguides. The guide is a dielectric rod of refractive index η immersed in another dielectric of slightly smaller refractive index n( l — Δ) ; both are in contact with a third dielectrio which may be air (Fig. la ) or a dielectric of refractive index n( l — Δ) , 2071
2072 THE BELL SYSTEM TECHNICAL JOURNAL,SEPTEMBER 1960 AIR n n(-△) n(-△) (a) (b) Fig.1-Dielectric waveguides for integrated optical circuitry. (Fig.1b).These geometries are attractive not only because of sim- plicity,precision of construction,and mechanical stability,but also because by choosing A small enough,single-mode operation can be achieved with transverse dimensions of the guide large compared with the free space wavelengths,thus relaxing the tolerance requirements. Even though in a real guide the cross section of the guiding rod is not exactly rectangular and the boundaries between dielectrics are not sharply defined,as in Fig.1,it is worth finding the characteristics of the modes in the idealized structure and the requirements to make it a single-mode waveguide. Furthermore,directional couplers made by bringing two of those guides close together,Fig.2,may become important circuit compo- nents..3 In this paper we study the transmission through such a coupler;the modes in a single guide result as a particular case,when the separation between the two guides is so large that the coupling is negligible.Through use of a perturbation technique,we also find the coupler properties when the two guides are slightly different. b n(t-4) n(1-A Fig.2-Directional couplers
2072 THE BELL SYSTEM TECHNICAL JOUBNAL, SEPTEMBER Ιθβθ AIR (a ) ( b ) Fig. 1 — Dielectric waveguide s for integrated optical circuitry. (Fig. lb). These geometries are attractive not only because of simplicity, precision of construction, and mechanical stability, but also because by choosing Δ small enough, single-mode operation can be achieved with transverse dimensions of the guide large compared with the free space wavelengths, thus relaxing the tolerance requirements. Even though in a real guide the cross section of the guiding rod is not exactly rectangular and the boundaries between dielectrics are not sharply defined, as in Fig. 1, it is worth finding the characteristics of the modes in the idealized structure and the requirements to make it a single-mode waveguide. Furthermore, directional couplers made by bringing two of those guides close together, Fig. 2, may become important circuit components.'*^ In this paper we study the transmission through such a coupler; the modes in a single guide result as a particular case, when the separation between the two guides is so large tha t the coupling is negligible. Through use of a perturbation technique, we also find the coupler properties when the two guides are slightly different. b 1 η \ η {.-Δ ) / Fig. 2 — Directional couplers
DIELECTRIC WAVEGUIDE 2073 The guiding properties of the rectangular cross section guide im- mersed in a single dielectric are compared with those derived through computer calculations by Goell.+Similarly,the coupling properties of two guides of square cross section immersed in a single dielectric are compared with those of two guides of circular cross section derived by Jones and by Bracey and others.3.In both comparisons agreement is quite good. II.FORMULATION OF THE BOUNDARY VALUE PROBLEM For analysis,we redraw in Fig.3 the cross section of the coupler subdivided in many areas.Nine of the areas have refractive indexes n to ns;we do not specify the refractive indexes in the six shaded areas.The reasons for these choices will become obvious. A rigorous solution to this boundary value problem requires a com- puter;nevertheless,it is possible to introduce a drastic simplification which enables one to get a closed form solution.This simplification arises from observing that,for well-guided modes,the field decays exponentially in regions 2,3,4,and 5;therefore,most of the power travels in regions 1,a small part travels in regions 2,3,4,and 5,and even less travels in the six shaded areas.Consequently,only a small error should be introduced into the calculation of fields in regions 1 if one does not properly match the fields along the edges of the shaded areas. The matching made only along the four sides of regions 1 can be achieved assuming simple field distribution.Thus the field components in regions 1 vary sinusoidally in the x and y direction;those in 2 and 4 vary sinusoidally along x and exponentially along y;and those in regions 3 and 5 vary sinusoidally along y and exponentially along r. The propagation constants k,k2,and ke along x in media 1,2,and 么 n 刀77 n Fig.3-Coupler cross section subdivided for analysis
DIELECTRIC WAVEGUIDE 2073 The guiding properties of the rectangular cross section guide immersed in a single dielectric are compared with those derived through computer calculations by Goell.* Similarly, the coupling properties of two guides of square cross section immersed in a single dielectric are compared with those of two guides of circular cross section derived by Jones and by Bracey and others.' " In both comparisons agreement is quite good. II. FORMULATION OF THE BOUNDARY VALUE PROBLEM For analysis, we redraw in Fig. 3 the cross section of the coupler subdivided in many areas. Nine of the areas have refractive indexes Πι to «β; we do not specify the refractive indexes in the six shaded areas. The reasons for these choices will become obvious. A rigorous solution to this boundary value problem requires a computer;*' nevertheless, it is possible to introduce a drastic simplification which enables one to get a closed form solution. This simplification arises from observing that, for well-guided modes, the field decays exponentially in regions 2, 3, 4, and 5; therefore, most of the power travels in regions 1, a small part travels in regions 2, 3, 4, and 5, and even less travels in the six shaded areas. Consequently, only a small error should be introduced into the calculation of fields in regions 1 if one does not properly match the fields along the edges of the shaded areas. The matching made only along the four sides of regions 1 can be achieved assuming simple field distribution. Thus the field components in regions 1 vary sinusoidally in the χ and y direction; those in 2 and 4 vary sinusoidally along χ and exponentially along y; and those in regions 3 and 5 vary sinusoidally along y and exponentially along x. The propagation constants fc,i, fc,o, and fc,« along χ in media 1, 2, and • I "a |b >^-a--Hf- - c 4^-a-*í^ Fig. 3 — Coupler cross section subdivided for analysis
2074 THE BELL SYSTEM TECHNICAL JOURNAL,SEPTEMBER 1969 4 are identical and independent of y.Similarly,the propagation con- stants k1,kys,and kys along y in the regions 1,3,and 5 are also identical and independent of z. In the appendix we calculate these propagation constants and find, as expected,that all the modes are hybrid and that guidance occurs because of total internal reflection.Nevertheless,because of another approximation which consists of choosing the refractive indexes na, na,n,and ns slightly smaller than n,total internal reflection occurs only when the plane wavelets that make a mode impinge on the inter- faces at grasing angles.*Consequently,the largest field components are perpendicular to the axis of propagation;the modes are essentially of the TEM kind and can be grouped in two families,Es,and E.The main field components of the members of the first family are E.and H,while those of the second are E,and H.The subindex p and g indicate the number of extrema of the electric or magnetic field in the x and y directions, respectively.Naturally,Ei:and Ei are the fundamental modes;we concentrate on them as we discuss the transmission properties of different structures. III.GUIDE IMMERSED IN SEVERAL DIELECTRICS The guide immersed in several dielectrics (Fig.4a)is derived from Fig.3 by choosing (1) It supports a discrete number of guided modes which we group in two families E andE plus a continuum of unguided modes. 8.1 The E Modes The main transverse field components of the modes are E,and H.. They are depicted in solid and broken lines,respectively,in Fig.4a for the fundamental mode Ei.Within the guiding rod each component varies sinusoidally both along x and along y.Outside the guide each component decays exponentially.Such functional dependence is given in equation(38)and depicted in Fig.4b.We assume n,≠n%≠n,≠ns; consequently the field distributions are not symmetric with respect to the planes x-0 and y=0.In Fig.5a we assume na=n and na =n; the E modes depicted are either symmetric or antisymmetric with respect to the same planes.These modes look similar to those in laser *This approximation is not very demanding.Even when m is 50 percent larger than ne,ne,n,and ns,the resulta are valid
2074 THE BELL SYSTEM TECHNICAL JOTTBNAL, SEPTEMBER IBSO * Thi e approximation is no t very demanding. Eve n whe n τΐι is 60 percent larger than fla, fu, tu, and tu, the resulte are valid. 4 are identical and independent of y. Similarly, the propagation constants kyi, kyt, and kyt along y in the regions 1, 3, and δ are also identical and independent of x. I n the appendix we calculate these propagation constants and find, as expected, tha t all the modes are hybrid and tha t guidance occurs because of total internal reflection. Nevertheless, because of another approximation which consists of choosing the refractive indexes n , , fit, n« , and slightly smaller tha n n, , total internal reflection occurs only when the plane wavelets tha t make a mode impinge on th e interfaces at grazing angles.* Consequently, the largest field components are perpendicular to the axis of propagation; the modes are essentially of the TEH kind and can be grouped in two families, El^ and El,. Th e main field components of the members of the first family are and H,, while those of th e second are E , and . The subindex ρ and g indicate the number of extrema of the electric or magnetic field in the χ and y directions, respectively. Naturally, El^ and Eli are the fundamental modes; we concentrate on them as we discuss the transmission properties of different structures. m. OOIDB IMMERSED IN SEVERAL DIELECTBICS The guide immersed in several dielectrics (Fig. 4a) is derived from Fig. 3 by choosing c = «. (1) It supports a discrete number of guided modes which we group in two families El, and El, plus a continuum of unguided modes.*'* 8.1 The El, Modes T h e main transverse field components of th e El, modes are E, and Ή,. The y are depicted in solid and broken lines, respectively, in Fig. 4a for t h e fundamental mode E\^. Within the guiding rod each component varies sinusoidally both along χ and along y. Outside th e guide each component decays exponentially. Such functional dependence is given in equation (38) and depicted in Fig. 4b. We assume n , ni n« ne ; consequently the field distributions are not symmetric with respect t o t h e planes χ - 0 and y = 0. I n Fig. 5a we assume nt = n« and η» = n, ; the El, modes depicted are either symmetric or antisjrmmetric with respect to the same planes. These modes look similar t o those in laser
DIELECTRIC WAVEGUIDE 2075 CONSTANT e-y/2 n2 光wi 2 cos(kyy+〕 Ey ns n3 777 227 OR Hz ELECTRIC FIELD CONSTANT e/ (b) 安 MAGNETIC FIELD Ey OR Ha cos (ka+a) CONSTANT e/作， CONSTANT e/ (a) Fig.4-Guide immersed in different dielectrics:(a)croes section and (b)field diatribution of the fundamental mode Bu". cavities with rectangular fat mirrors,but our nomenclature is different. The subindexes p and g indicate the number of extrema each component has within the guide. Now we describe these modes quantitatively by reproducing the propagation constants found for each medium in Section A.1 of the appendix.Let us call k.the axial propagation constant and ka,and k the transverse propagation constants along the z and the y directions, respectively,in the wth medium(=1,2,...5).Furthermore,let us call k=a,-笑 (2) the propagation constant of a plane wave in a medium of refractive index n,and free-space wavelength A. According to equations(39)through(52) k,=(一2-幼 (③)
DIBLECTBIC WAVBQOIDE 2075 ttt ELECTRIC FIELD CONSTANT e'/ts \ \ CONSTANT e"*/ti Fig. 4 — Guide immersed in different dielectrics: (a) cross section and (b) field distribution of the fimdamental mod e Eu'. cavities with rectangular flat mirrors, but our nomenclature is different." T h e subindexes ρ and g indicate the number of extrema each component has within the guide. Now we describe these modes quantitatively by reproducing the propagation constants found for each medium in Section A.l of th e appendix. Let us call k, the axial propagation constant and fc„ and fc„ the transverse propagation constants along the χ and the y directions, respectively, in the vth medium (» = 1, 2, · · · 5). Furthermore, let us call k. 1^ 2» «n, = — n . (2) the propagation constant of a plane wave in a medium of refractive index n, and free-space wavelength λ. According to equations (39) through (52) k. = (fc? -kl - k^* (3)
2076 THE BELL SYSTEM TECHNICAL JOURNAL,SEPTEMBER 1969 Ey OR Ha ↑Ey OR Ha cos kz CONSTANT e-2/传 CONSTANT e-yA cos kyy Ey OR Hz E29 4↑40↓+ h+10 Ey OR Ha (a) E OR Hy Ee OR Hy cos ka CONSTANT e-2/E CONSTANT e-y/ cos kyy E Ex OR Hy E Ez OR Hy (b) ELECTRIC FIELD --◆MAGNETIC FIELD Fig.5-(a)Field configuration of E"modes.(b)Field configuration of E modes. in which ks ka=k2=k (④) and k,=k1=ka=k8… (5) This means that the fields in media 1,2,and 4 have the same x
2076 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 196» COS kj, i Eu OR H j CONSTANT e Ey OR Hj T " b «—a— » . t . . 1 . • t . .1 . . j . .ft (a) CONSTANT e- / OR R «A c O R H b COS kxX . EG O R Hy CONSTANT e-*/< Ee OR Hy CONSJANT e-y/ ' y ^co s ky y / OR H „ - c -12 -22 O R H l ( b ) • ELECTRIC FIELD • MAGNETIC FIELD Fig. 5 — (a) Field configuratíon of modes, (b) Field configuration of E„' modes. in which and (4) fc, = A;,i — Ä.s — fc,e · (5) This means tha t the fields in media 1, 2, and 4 have the same χ
DIELECTRIC WAVEGUIDE 2077 dependence and similarly those in media 1,3,and 5 have identical y dependence.These transverse propagation constants are solutions of the transcendental equations: k,a pr -tan-1 kta -tankts (6) ,b=g红-tan1生k,n-tan, (7) 1 in which (8) 4 (9) and A.8=-4.=20m-.. (10) In the transcendental equations (6)and (7),a and b are the trans- verse dimensions of the guiding rod,and the tan-1 functions are to be taken in the first quadrant. What are the physical meanings of,,and A...?The amplitude of each field component in medium 3(Fig.4)decreases exponentially along z.It decays by 1/e in a distance=1/k l.Similarly,, andn measure the "penetration depths"of the field components in media 5,2,and 4,respectively. The meaning of Aa is the following.Consider a symmetric slab derived from Fig.4 by choosing a oo and na n.The maximum thickness for which the slab supports only the fundamental mode is A,. Expressions(3),(8),and (9)contain k,and k,,which are solutions of the transcendental equations(6)and(7).These cannot be solved exactly in closed form.Nevertheless,for well-guided modes,most of the power travels within medium 1,implying k:As <1 and K1. (11) It is possible then to solve those transcendental equations in closed
DIELECTRIC WAVBGXnD E 2077 dependence and similarly those in media 1, 3, and 5 have identical y dependence. These transverse propagation constants are solutions of the transcendental equations: in which k,a = ρπ - tan ' ^,{ 3 — tan ' k,^t fc,6 = q-ir - tan" -'^Μ ' - | . -tan->^fc., . n, (6) (7) s V2 * « x 3 S 2 'i r - k l 5 k,2 π A, L I 4J - K (8) (9) and A a ,3,4 ,s — (10) (fc ? - fc».3...5)* " 2R^^"3 | . Similarly ξ», v» 1 and 174 measure the "penetration depths " of the field components in media 5, 2, and 4, respectively. T h e meaning of A] is the following. Consider a symmetric slab derived from Fig. 4 by choosing a = » and nj = n« . The maximum thickness for which the slab supports only the fundamental mode is A,. Expressions (3), (8), and (9) contain fc, and fc, , which are solutions of the transcendental equations (6) and (7). These cannot be solved exactly in closed form. Nevertheless, for well-guided modes, most of the power travels \vithin medium 1, implying k,A « 1 and k,A « 1. (11) It is possible then to solve those transcendental equations in closed
2078 THE BELL SYSTEM TECHNICAL JOURNAL,BEPTEMBER 1969 though approximate,form.Their solutions are =(+4t4 (12) =答(+贴+4 xn b (13) For large a and b,the electrical width,ka,and the electrical height, k,b,of the guide are close to pa and g,respectively. Substituting equations(12)and (13)in equations(3),(8),and (9), we obtain explicit expressions for k=,s,s,n,and: =[-(+4结4-(gl+4门 xnib (14) + (15) (16) 32 The Ei Modes Except for the fact that the main transverse components are E.and H.,the E modes are qualitatively similar to the E modes (Fig.5b); they differ quantitatively.Distinguishing with bold-face type thesymbols corresponding to E modes,the axial propagation constant and the "penetration depth"in media 2,3,4,and 5 are,according to equations (60),(63),and(64), k,=(好-经-) (17) (18) 南因 (19)
2078 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 196» a \ πα / κ πα ηΐΑ, + rri[b l·nlAλ-' (12) (13) For large a and b, the electrical width, k^a, and the electrical height, kyb, of the guide are close to ρπ and qw, respectively. Substituting equations (12) and (13) inequations (3), (8), and (9), we obtain explicit expressions for fc«, es, is, i?2, and ij4: k. = As + A ira ·Γ-(?)'(-=^Γ] ' (14) A, 9 Τ A, 1 - 1 - ρΑϊ 5 . α ^ _|_ A3 + A5 ira gAj 4 b n|A^jfnjA . 1 Η —:5t; (15) (16) 3Λ TAe .β;. Modes Except for the fact tha t the main transverse components are and H, , the E'„ modes are qualitatively similar to the .EJ, modes (Fig. 5b); they differ quantitatively. Distinguishing with bold-face type the symbols corresponding to E'„ modes, the axial propagation constant and the "penetration depth" in media 2, 3, 4, and 5 are, according to equations (60), (63), and (64), k . = {k\ - - «3 = 5 nj 4 k k . , 4 (17) (18) (19) though approximate, form. Their solutions are
DIELECTRIC WAVEGUIDE 2079 in which k.and k,are solutions of the transcendental equations ka=pm-tan4光k,5-tan1元k6 (20) k,b gx-tan"k,ma -tan"ik,n. (21) The approximate closed form solutions of these equations are k,=(+贴4+4) xnja (22) and -+4者4 (23) Substituting these expressions in equations (17),(18),and (19),we derive the explicit results: -[-(l+44-(gl+44] (24) 在-1-。，+ (25) A 94 61+4+4 1 1- (26) If 《1, these results coincide with those in equations (14),(15),and (16), indicating that the E3 and E modes become degenerate. 8.3 Examples The axial propagation constants k.and k.,given in equations(3) and(17)and properly normalized,have been plotted in Figs.6a through k as a function of the normalized height of the guide 名=尖阳-
DIELECTRIC WAVEGUIDE 2079 in which k. and k, are solutions of the transcendental equations t o = pir - tan" ' ^k . ? s - tan" ' ^k,e . k,6 = ς* — tan" ' k^nj — tan" ' k^n« . The approximate closed form solutions of these equations are and a \ vn'a / (20) (21) (22) (23) Substituting these expressions in equations (17), (18), and (19), we derive the explicit results: (24) γ γ \ ^ (25) k. = A, A^ 1 - 1 - pAs s α J _|_ nlAi + nlAt — πη,α qAi 4 1 6 1 , A, + A, 1-J (26) If n, I 4 5 « 1 , these results coincide with those in equations (14), (15), and (16), indicating tha t the E'^ and E',, modes become degenerate. 3 .3 Examples The axial propagation constants k, and k., given in equations (3) and (17) and properly normalized, have been plotted in Figs. 6a through k as a function of the normalized height of th e guide b _2b ,2 „a^J
2080 THE BELL SYSTEM TECHNICAL JOURNAL,SEPTEMBER 1969 1.2 n4 1.0 E,Ei2 E、En9、 0.8 1.05 <n4<n E2,E2IV e,E22 0.6 w1 E,E1 0.4 E3,Ear 也（起 0.2 E,E29 (a) e,e327 E5,E3 n4 E开，En 1.0 E2i,E2V 105 <n4<n E3,E3. 0.8 信=2 0.8 =n E话，E2 0.4 -E五，e22 0.2 (b) 0 04 08 1.2 1.6 2.0 2.4 2.8 3.2 3.8 4.0 贵=驶(6h-n） Fig.6-Propagation constant for different modes and guides. tran- scendental equation solutions;- 一clo8 ed form solutions;一·-·一Goell's computer solutions of the boundary value problem. for several geometries and surrounding media.*The ordinate in each of these figures is 发=当 it varies between 0 and 1.It is 0 when k.=k,that is,when the guide +In these figures we use the same symbol k,for both the E and the E modes
2080 THE BELL SYSTEM TECHNICAL JOUBNAL, SEPTEMBER 1969 I.Zr Fig. 6 — Propagation constant for different mode s and guides. transcendental equation solutions; closed form solutions; — Goell's compute r solutions of the boundary value problem. for several geometries and surrounding media.* The ordinate in each of these figures is fc^ - kl it varies between 0 and 1. It is 0 when k, = kt, tha t is, when the guide • I n these figures we use the same symbo l k, for both the E,,' and the E„' modes