Kato, Electron. Lett. 13, 53 (1977).7 R. K. Jain, C. Lin, R. H. Stolen and A. Ashkin, Appl. Phys. Lett.

31, 89 (1977).8 C. Lin, R. H. Stolen, W. G. French, and T. G. Malone, Opt. Lett.

1, 96 (1977).9 R. H. Stolen, C. Lin, and R. K. Jain, Appl. Phys. Lett. 30, 340

(1977).0 C. Lin, R. H. Stolen, and L. G. Cohen, Appl. Phys. Lett. 31, 97

(1977).

"L. G. Cohen and C. Lin, Appl. Opt. 16,3136 (1977).12 R. G. Smith, Appl. Opt. 11, 2489 (1972).13 F. Capasso and P. D. Porto, J. Appl. Phys. 47, 1472 (1976).14 J. AuYeung and A. Yariv, IEEE J. Quantum Electron. QE-14, 347

(1978).15 D. C. Johnson, K. 0. Hill and B. S. Kawasaki, Radio Sci. 12, 519

(1977).1 A. Yariv, Quantum Electronics, 2nd ed. (New York, Wiley,

1975).

Analysis of optical channel waveguides and directionalcouplers with graded-index profile

Toshiaki Suhara, Yuichi Handa, Hiroshi Nishihara, and Jiro KoyamaOsaka University, Department of Electronics, Faculty of Engineering, Yamada-kami, Suita, Osaka 565, Japan

(Received 28 December 1978)

A theoretical analysis of graded-index optical channel waveguides and directional couplers fab-ricated by the beam-writing technique is presented. The analysis is based on a combination of theeffective-index-of-refraction method and the WKB method, which is shown to apply for directionalcouplers as well as for channel waveguides, when special care is taken in connecting the field at thegap region. The mode dispersion characteristics and the field profile of a channel waveguide werederived. The conversion length of a directional coupler was also determined as a function of thecoupler parameters and the mode order. The guided mode field has long evanescent tails outside theguide, and the positions of the turning points are located further away from the guide center forhigher-order modes. Accordingly, the conversion length of a coupler increases slowly with the wave-guide spacing and exhibits a strong dependence on the transverse mode order. The results aresummarized in a few practically useful design charts by employing the normalizations of the deviceparameters. Some design examples are given and the fabrication tolerances are discussed.

1. INTRODUCTION

Optical channel waveguides and directional couplers areamong the most important elements in constructing inte-grated optical circuits and have been actively studied. Inparticular, devices with a step refractive-index profile fabri-cated by lithography and etching technique have been ex-tensively analyzed."12 The fundamental properties are wellunderstood and the design techniques are established. Anumber of experimental works have been performed aswell. 3-5

On the other hand, the recent developments in waveguidematerials and fabrication techniques have made it possibleto produce channel waveguides and directional couplers witha graded refractive-index profile. There have been great in-terests in such devices because of their excellent waveguidingcharacteristics, easy fabrication and advantages in givingvarious additional functions. However, optical devices witha graded index profile are quite difficult to analyze. For thisreason, they used to be designed by employing an approxi-mation to a step index profile. An accurate analysis for thegraded index profile remains to be developed.

A channel waveguide fabricated by the selective diffusionof metal in an electro-optical crystal has an index profilegraded mainly in the direction of the substrate thickness. Thewave-guiding mode characteristics were analyzed by meansof the effective-index-of-refraction method by Hocker et al. 6

Taylor employed a numerical method to analyze the mode

dispersion characteristics taking into account the waveguidebroadening by the transverse diffusion.7 As for a directionalcoupler, Taniuchi et al. applied the multiple layer decompo-sition method to a two-dimensional model and compared thecoupling characteristics of couplers with graded and step indexprofiles.8 However, detailed analysis giving a guide for de-signing couplers with a graded three-dimensional index profileseems not yet to have been reported.

The authors have developed a direct electron-beam writingtechnique for the fabrication of micro-optical components,making use of the electron-beam-induced refractive-indexincrease in chalcogenide amorphous semiconductor thinfilms.9-' 2 By this technique, one can fabricate channelwaveguides and directional couplers with an index profilegraded in the transverse direction along the plane of thewaveguide. Channel guides and couplers with similar indexprofile were made by the laser beam writing technique inphotosensitive waveguide materials.13'1 4 The design theoryfor this type of device, however, is not well established, anda considerable mismatch has been reported between the ex-perimentally obtained coupling characteristics and thosepredicted theoretically by using the step index approxima-tion.14

The purpose of this paper is to analyze optical channelwaveguides and directional couplers with graded index profilefabricated by the electron- or laser-beam writing techniqueand establish the design procedure. In the following sections,we first show that the devices can be analyzed by a combina-

807 J. Opt. Soc. Am., Vol. 69, No. 6, June 1979 0030-3941/79/060807-09$00.50 (O 1979 Optical Society of America 807

ELECTRON BEAMCHANNEL (LASER BEAM)WAVEGUIDEl

WVE=GIDNLAYER

FIG. 1. Schematic illustration of the fabrication of a channel waveguideby the bean writing technique.

tion of the effective-index-of-refraction method and the WKBmethod. We then clarify the characteristics peculiar to thegraded index type. Finally, we give some design examples anddiscuss the results obtained in comparison with devices havinga step index profile.

11. CHANNEL WAVEGUIDE

A. ModelConsider a channel waveguide fabricated in a thin film by

the electron- or laser-beam writing technique as illustratedin Fig. 1. The cross section of the guide is depicted in Fig. 2.The refractive index is graded in the x direction and varies insteps in the y direction. The profile is shown by the solid linesin Fig. 3 and expressed by

ni, T<y,

n(x,y) n20 + An2f(x/a), 0 < y < T, (1)

n3 , y< O

where T is the thickness of the waveguiding layer, and nI, n2 0 ,

and n3 are the refractive indices of the air, the guiding layer,and the substrate, respectively. The function f(x/o), de-scribing the profile in the transverse x direction of the indexincrement that performs the channel guide, has a maximumvalue 1 at x = 0 and satisfies f(-x/cr) = f(x/cr) and f(+±c) =0. The waveguide width 2cr need not necessarily satisfy 2ar>> T.

We assume for simplicity that the refractive-index incre-ment An2 at the guiding region is small, so that

An2 << n2O. (2)

Then the propagation constants of the waveguide modes areconfined within a narrow range of values and the derivativeof the field amplitude with respect to the x direction is muchsmaller than those with respect to the y and z directions.Thus the following rather formal relations hold:

n. (x)

FIG. 3. Distribution of the re-fractive index in the channelwaveguide.

(3)|1 2I 1, Ik 2 I2H<" ,

where k = cO(AEO)1/ 2 is the wave number in free space.

The authors' preliminary experiment showed that, in thecase of a fabrication by electron-beam writing, the indexprofile is a Gaussian type, which reflects the current densitydistribution of an electron beam, and f(xlo) can be writtenas1ol1 :

f(x/cr) = exp[-(x/j)2j. (4)

It would be reasonable to assume the same profile for awaveguide fabricated by laser-beam writing. Therefore wewill use Eq. (4) for calculation in the following sections, al-though the analysis is generally applicable to arbitrary indexprofiles. The time and z dependences of the field are assumedto be expjj(wt - 13z)], where d is the propagation constant, andwill be omitted.

B. Wave equations and boundary conditionsThe electromagnetic field of guided light in the waveguide

shown in Fig. 2 or 3 must satisfy Maxwell's equations and thefollowing wave equations:

V2 E + V[E - Vn 2/n 21 + k2 n2 E = 0,

V2H + (Vn2/n 2 ) X (V X H) + k2 n2H = 0.

(5a)

(5b)

The guided modes are hybrid due to the graded variation inthe x direction and the discontinuity in the y direction of therefractive index, and are classified into EX and Ey modes.

1. Ex mode (E almost parallel to the x axis)Put Hx = 0, then through the Maxwell's equations the other

field components are described in terms of H, as

F ~ I~~!H iaia (Hy 1jf ay , EXf d IY k2ax n2 ax )

1 a 2Hy E j aHyO=fEoOf2 bxy z = e(n2 ax

Making use of the relation Eq. (3), we find from Eq. (6)

Hy ne(f3/)Ex,

yAir n

T

Guiding Z 0layer

Substrate n3.

FIG. 2. Cross section of achannel waveguide.

(6)

(7)

and we therefore can consider E. as an independent fieldcomponent instead of Hy. The wave equation which Ex mustsatisfy is derived from Eq. (6a) to be

a2 a 2 a ( Ian 2 E,-E +---Ex+-- I--Eax2 bya2 b ox \n2 ax I

+ (k2 n 2 - I 2)Ex = 0, (8)

and is rewritten approximately under the condition of Eq. (3)

808 J. Opt. Soc. Am., Vol. 69, No. 6, June 1979 Suhara et al. 808

2.5

(2 2 2-I-+-+ (k~n2 -R 2)IE. = 0.

\0x 2 ay 2 rj (9)

On the boundary planes at y = 0 and T, the tangentialcomponents of the field must be continuous. This is auto-matically satisfied for Hx since Hx = 0. The condition for E,can be omitted, since a relation E, I << E., I holds from Eqs.(3) and (6). Thus the boundary conditions for Ex mode arewritten as

Ex = continuous, = continuous.(y

(10)

2. EY mode (E almost parallel to the y axis)Put E, = 0, then the other field components are described

in terms of Ey as

. E = , ,y = - coon (1 + I!2nt EYE, = - 3 YXH. = -

(3 ay, ' 3 h2n2ax2Y

Hy,= - .13 H;,= y

Making use of Eq. (3), we find from Eq. (11)

E, --(/31wcn2)H,. (0.2)

This allows considering H.x as an independent field compo-nent. The wave equation for Hx is derived from Eq. (5b) tobe

12 + (2 + (k 2n2 -/3 2)I H = 0. (13)\0X 2 0y 2/

In a similar way as for the Ex mode, the boundary conditionsfor the BY mode are written as

Hx = continuous, (1/n 2)(oH,/by) = continuous. (14)

C. The effective-index-of-refraction methodBy the discussion in Sec. IIB, the problem of knowing the

electromagnetic fields of EX and EY modes was reduced to thatof solving a wave equation

)a2 + + [k2n2 (Xy) - /32]) G(xy) = 0, (15)

G (xVy)= Ex (xXy) Ex mode, (16){HEx(x,y), EYmode,

under the boundary conditions given by Eq. (10) or (14) at y= 0 and T. We apply the effective-index-of-refractionmethod to Eq. (15) by putting

G(x,y) = G(x) G(y). (17)

Then we have the following separated wave equations:

(dx2 + 2ne (x) - ,2]) G(x) = 0, (18)

[d2 + k2 [n4(y) - n2]) G(y) = 0, (19)

where

n2(x) = n + 2n 2oAn2 (ftx/c) - 11, (20)

xN0z

H

HE-

C)Q

0 0.5 1.0FILM THICKNESS T/X

FIG. 4. Mode dispersion curves for determining the effective index ofrefraction with y-mode order n.

ni I T<y,

no(y) = n2 = n2( + An 2 , 0 < y < T,

n3 , y <0,

(21)

.1) and ne is the effective index for the nth mode with respect tothe y direction. The boundary conditions Eqs. (10) and (14)are rewritten in terms of G(y) as

Ex mode: G(y), dG(y)Idy= continuous at y = 0, T, (22)

EYmode: G(y), [no(y)j]2dG(y)/dy= continuous at y = 0, T. (23)

Solving the wave equation by the effective-index-of-re-fraction method is equivalent to deriving the field by modi-fying the index distribution in the air and the substrate regionsas shown by the broken curves in Fig. 3. The field amplitudein the regions where the index distribution is modified, how-ever, is so small that our approximate treatment may resultin good accuracy of calculation. This asserts that Eq. (21) isthe most proper choice for the refractive index in the imagi-nary slab waveguide, for such a choice minimizes the effect ofthe index modification. In fact, the authors' estimation bymeans of the variational methodl5 showed that the possibleerror in the calculated value of the normalized propagationconstant b, which we define in Sec. II D, is less than 3% exceptfor the modes near cutoff, and their accuracy is adequate forpractical design purposes.

Equation (19) and Eqs. (22) and (23) are no more than thewave equation and the boundary conditions for TE and TMmodes for a slab waveguide with a step index distributionno (y). As found in the literature, the effective index ne andthe y dependence of the field G(y) can readily be obtained.We therefore show in Fig. 4 only the calculated result of ne forni = 1.00 (air), n2 = 2.47 (As 2SO) and n 3 = 1.54 (Corning# 7059 glass) for convenience of later practical designing.

D. Mode dispersion characteristics and field profileWe here normalize the x coordinate into u by

X = 0-U,

and define the following parameters:

V = hVJa,

(24)

(25a)

809 J. Opt. Soc. Am., Vol. 69, No. 6, June 1979 Suhara et al. 809

(25b) 1.

(26a)

(26b)

The parameter V is the normalized frequency or the nor-malized waveguide width and b is the normalized propagationconstant satisfying 0 < b < 1. Then the wave equation, Eq.(18) related to the x direction, is reduced to

(Id2 + V 2[f(u) - b]) G(x) = 0. (27)

The solutions of Eq. (27) satisfying G (+G ) = 0 can be obtainedto a good approximation by employing the WKB method.'6

The field profile G(x) exhibits oscillatory and exponentiallydecaying behaviors inside and outside the turning points (x= ±Out), respectively. The normalized coordinate ut (>0)of the turning points is determined by

f(ut) - b = 0. (28)

The function G (x) must be an even or odd function of x sincef(-u) = f(u), and therefore the derivation of G(x) in x > 0 issufficient. Letting

k(u) = Vjf(u) - b6l/2, 0 • U < Ut, (29a)

K(U) = VJb - f(U)I/2It < U,

VW) = f J k(u) du, 0 < u < ut,fu

l(u) = K(u) du, Ut < i,at

one can write the solution of Eq. (27) as

G(x) =

a = 2n 204n 2,

b = [(P/k)2 - n2d]/a,

ned = n,-a.

O//k = (ab + ned) -2 ned + ab/2ned, (33)

where ned is given by Eq. (26b) with the nth effective indexne for Ex (EY) mode. Figure 5 shows the relation between band V calculated from Eq. (32) for the Gaussian index profilegiven by Eq. (4).

(29b) The highest order M of the transverse x mode in a givenwaveguide is also determined from Eq. (32). The left-hand

(30a) side of Eq. (32) is a monotonically decreasing function of b andtakes a positive value less than (r/2)1/2V. Therefore we see

(30b) that M is the largest integer such thatM + 1/2 < (2/ir) 1/

2 V, (34)

and the criterion for a waveguide to support a single x modeis

A[k(u)]'1/2

cos{(vu) - "

A (2-7r3(u) )1/2 VIA(u)] + J-jys[1(u)11

4 (2ir(U))1/2 I- 13[n(u)] + I-1/3[W(u)]

A [K(u)>-1/ 2 exp[-q7(u)l2

0< iU <Ut,

, U < ut,

, ut US , (31)

U ut < U,

where A is a constant and J,, and I, denote the Bessel andmodified Bessel functions of order v, respectively. Here usehas been made of the asymptotic expansions of J, and I,, toconnect the solution at the turning points. The characteristicequation related to the x direction is

VfV(u) -b]1/ 2 du= (m + )1' (mn = 0,1, 2,. .. ).

(32)

The function G(x) is even or odd according to whether themode order m is even or odd.

It should be noted that the characteristic Eq. (32) is com-mon to both EX and Ey modes, and is described by using onlythe normalized frequency V and the normalized propagationconstant b, regardless of the y-mode order n. It is apparentfrom Eq. (26) that, when b is determined as a root of Eq. (32)with given m, the propagation constant j3 or the mode index

lf2r1/2 < V < 3 (Ir/ 2

2 2 2 2(35)

The field profiles in the transverse x direction, G (x), cal-culated for V = 1.0,1.5 (single-mode) and 5.0 (multimode)guides, are plotted in Figs. 6(a), (b), and (c), respectively. Thefull field profile of the Exn (Eyn) mode is visualized by theproduct of Gm (x) and G, (y), where G,, (y) is the profile of thenth EX (EY) mode in the imaginary slab guide. The x profilesG (x) are characterized by an evanescent tail which decays veryslowly from the turning points. Comparison between Figs.6(a) and (b) shows that the turning points move toward thecenter of the guide for larger V, indicating the stronger fieldconfinement. We also see from Fig. 6(c) that the higher theorder of the guided mode in a given waveguide, the furtherfrom the center of the guide the turning points locate. Thesewaveguiding characteristics are ones peculiar to the gradedindex type and account for the coupling characteristics of thedirectional coupler to be clarified in the next section.

Ill. DIRECTIONAL COUPLER

A. ModelConsider a directional coupler constructed by writing two

identical waveguides parallel with a center-to-center spacing2d. Figure 7 illustrates the cross section of the coupler. Therefractive-index distribution is shown in Fig. 8 and is ex-pressed by

810 J. Opt. Soc. Am., Vol. 69, No. 6, June 1979

0Mn 0

00.52

0 0

NORMALIZED FREQUENCY V

FIG. 5. Mode dispersion curves related to the transverse x direction. mdenotes the x-mode order and the curves are applicable to both Ex and Eymodes.

3/k for the Etn (Ey n) mode can readily be evaluated by

Suhara et al. 810

(a) v=1 .00G (x)

AIR

m= 0

-4 -3 -2 -1 0 1 2 3x/a

G (x)(b) V=1. 50

m0

I I 0 1-4 -3 -2 -1 0 1 2 3 ,_ 4

xKa

y

2a 20-e 3 To 4

GUIDING .,' 0 -n

GUIDING I 0 'LAYER k_2d

SUBSTRATE I

ni

n 3

FIG. 7. Cross section of the directional coupler.

metrical and antisymmetrical modes are closely equal tosymmetrical and antisymmetrical superpositions, respectively,of guided mode field distributions associated with each singlewaveguide. Here we put

(c) V=5. 00

2 3 4x/a

FIG. 6. Field distributions in the transverse x direction in channel wave-guides. The short vertical bars on the curves indicate the position of theturning points.

nj , T<y,

n(x,y) = 2 + An 2 g(xkof), 0 < y < T, (36)

n3 , y < 0,

and

g(x/f) = f[(x + d)/o-i + fE(x - d)la]. (37)

The function fAx/u) describes the index profile of a singlewaveguide and is given by Eq. (4) for the Gaussian profile.

B. Field distributionThe guided light field in the coupler shown in Fig. 7 can be

derived in a similar way as in the last section by means of theeffective-index-of-refraction method. In particular, the ydependence G(y) of the field can be obtained from Eq. (27).The WKKB method can also be employed to derive the x de-pendence G (x) by solving the resultant wave equation

(d2 + V 2[g(U) -b)G = 0. (38)

In the directional coupler the guided field has two turningpoints in x < 0 and two in x > 0. Their normalized coordi-nates are determined by the equation

g(U) - b = 0, (39)

and are denoted by huti and +Ut2 (0 < Utl < Ut2).

The guided mode in the coupling region is either a sym-metrical mode satisfying G(-x) = G(x) or an antisymmetricalmode satisfying G(-x) =-G(x) since the refractive-indexprofile is symmetrical. The field distributions of the sym-

k(u) = V[g(u) - b1"2 , Ut <u < Ut2,

K(U) = V[b - g(U)]112

, 0 < U < Uti, Ut2 < U,

and

Wu)= i U k (u) du, 0 < U < Ut2,2

{lU) 5 r Au) du ut, < > < ut2,

2( = K(U) du, Ut2 < u.Ut2

Then the solution of Eq. (38) can be written as

G(x) =

B 2K(U)}h"12expn-X 2 (u)l, Ut2<U, (42a)

2B 32(U) 1/ -IIl/[/i2(u) + I-1is[i2 (u)ZI, Ut2 < u, (42b)

-2rk(u 12J 1R3[A2 (u)] + J- 1/34 2 (u)]1, u < ut 2, (42c)

B[k(u)]- 1 /2 Cos[42 (u) - 7r/4], Utl < U < Ut2, (42d)

(~~)1/2¾k(u)) 1CJ13R[tI(u)] + DJ-1 3[41 (u)J}, ut l < U, (42e)

| nl(U) 1/2 {-C1113NIMu] + DI_1/3[n1(U)]1,

0 < u < utl, (42f)

symmetrical or antisymmetrical, u < 0, (42g)

n (x)

An2

.,n2

fl39. v

n~ 3--1 i

ni I

I I-Id ° d a- x

FIG. 8. Distribution of the refractive index in a directional coupler.

811 J. Opt. Soc. Am., Vol. 69, No. 6, June 1979

(40a)

(40b)

(41a)

(41b)

(41c)

(41d)

4

Suhara et al. 811

where B, C, and D are constants. The solution G (x) requiresthe continuity of G(x) and dG(x)/dx for all values of x.Equation (42) implies that they are connected at the outerturning points (u = lUt 2 ); the connection is based on the as-ymptotic expansions of the Bessel and modified Besselfunctions, which were used in Sec. II C to match the solutionfor a single waveguide. On the other hand, it is important tonote that in the gap region (I u I < ut i), the expression of G (x)by the modified Bessel functions should be retained. Thisis because, in the gap region, G(x) is a combination ofIl,3 [AnI(u)] and 1-1/3[HIM(u)] with different coefficients (C id

D) and therefore the approximation by the asymptotic ex-pansion would not be valid.' 7 The connection of G(x) at x= 0 with attention to this situation leads to the followingrelations between C and D.

Symmetrical mode: It is evident that, for a symmetricalfield profile G (x) [G (x) = G (-x)], the continuity of G (x) anddG(x)/dx at x = 0 is satisfied only when dG(x)/dx vanishesat x = 0 (u = 0). Thus we have

D I1/3(7) + 77[14/3(70) + I-23(1)] (43)

C I-1/3 (q) + W7(I-4/3(7n) + 12/3(70)1

where

v = 1(O) = V I b-g(u)j1/2 du (44)

is a measure of the waveguide spacing.

Antisymmetrical mode: For an antisymmetrical mode, thecontinuity of G (x) and dG (x)Idx is satisfied only when G (x)vanishes at x = 0 (u = 0). Thus we have

DIC = I11/3(77)/1I-13(7). (45)

C. Characteristic equationWe here derive the characteristic equation, which deter-

mines the propagation constants of the guided mode in thecoupler, by connecting the field at the inner turning points (u= +Uti). Making an asymptotic expression of Eq. (42e) andmatching it to Eq. (42d) yields a relation

B = + [2(C2 + CD + D2 )/ir]1/2, (46)

where the plus and minus signs stand for the symmetrical andantisymmetrical modes, respectively, and the characteristicequation is

V f5"' fg(u) - b]1/2 du + 6

ebl

= (m+ 2-X, (m=0,1,2, ... ), (47)

where

6 = tan[(D - C)/v3(C + D)],

symmetrical mode:

Is = tan-' ;

X 11/3 - -1/3 + 7[(14/3 - 1-4/3) - (12/3 - 1-2/3)11 4911/3 + I-1/3 + 7[(I4/3 + I-4/3) + (12,3 + I-2/3) 1

and combining Eqs. (48) and (45) yields an expression for theantisymmetrical mode:

(, = tan' I-1/2 (50)

where I = I, (ij). The dependences of 6, and 6a upon n arereadily outlined if we approximate, for the moment, Eqs. (49)and (50) by using the asymptotic expansion as

6, 1 tan-' [ (27- ) exp(-277)]

n tani (I exp(-2q))

6a !-tan-' [1/2 exp(-2-q)].

(49')

(50')

These formulas indicate that 6, and 6a take positive andnegative values, respectively, and their magnitudes, nearlyequal to each other, become small exponentially with in-creasing 77 corresponding to large waveguide spacing. Theapproximate expressions Eqs. (49') and (50'), however, are notaccurate enough to apply for general cases and therefore weuse the rigorous Eqs. (49) and (50) for calculation.

The pertinent problem was thus reduced to solving Eq. (47)with Eq. (44) and Eq. (49) or (50) for given V, d/v, and m.However, this requires a time consuming computer programsince the unknown b appears in the expressions for UtI, Ut2,

and 6 as well as in the integrand of Eq. (47). But, under theassumption of weak coupling, we can simplify the calculationby separating the problem into the field confinement in awaveguide and the effect of the mutual coupling between twoguides on the propagation constant. To do this, we first de-fine bo by

V fu" [g(u) - bo]l/ 2 du = (m + I/2)r..Uel

(51)

The root bo of Eq. (51) is the normalized propagation constantwith no coupling present. The roots of Eq. (47), b, and ba forthe symmetrical and antisymmetrical modes, respectively, areclosely equal to bo, provided that the coupling is weak. Thisallows a Taylor expansion of the integral term in the left-handside of Eq. (47) around b = bo. Leaving the leading term andcombining with Eq. (51) yields an approximate but explicitexpression for b, and ba:

bsa = bo + X 6 s,a,

X = V f t'jg(u) - bohi/2 du,2 at(48)

(52)

(53)

and m is the mode order related to the transverse x direc-tion.

The parameter 6 represents the phase shift at the innerturning points due to the mutual coupling between the twoguides and is expressed by different formulas for the sym-metrical and antisymmetrical modes. Eliminating C and Dfrom Eq. (48) by using Eq. (43) yields an expression for the

where the phase shift 6 and the coefficient X can be evaluatedapproximately by putting b = bo in their formulas. In thisway, b, and ba are obtained by simple integration instead ofby solving the complicated integral equation.

The practical evaluation of b, and ba was first, made bysolving Eq. (47) numerically. The results for the coupling ofthe fundamental (m = 0) mode in V = 1.0, 1.2, 1.5, and 2.0

812 J. Opt. Soc. Am., Vol. 69, No. 6, June 1979 Suhara et al. 812

1 l. . * * * ontns

Z0H

54U

0.014

0.501

00

Zo E0H Z

.4 E

E uZ

0 0ZU 0

0 l 2 3 4 5 6 7NORMALIZED WAVEGUIDE SPACING d/a

FIG. 9. Normalized propagation constants b2 and ba for the symmetricaland antisymmetrical modes vs. normalized waveguide spacing d/i for thecoupling of fundamental-mode (E'n and Eyo) fields in directional cou-plers.

waveguides are summarized in Fig. 9, wherein the hatchedregion indicates the range of values for dia where the innerturning points disappear and the structure supports guidedlight as a single waveguide. Note that the curves for b, havea maximum at a value of dia, while those for ba exhibit amonotonic behavior. In the left sides of the maxima the re-sults are not very accurate; the condition of the linear turningpoints assumed in our WKB analysis is violated near theboundary between the hatched and unhatched regions, onwhich the inner turning points transfer to a quadratic turningpoint. The comparison between the results obtained fromthe rigorous calculation and the approximate one using Eq.(52) showed that the error in b - bo yielded by the latter is lessthan 1% for dl > 2.5.

D. Conversion length and maximum power transferIn the foregoing discussion, two waveguides constructing

the coupler were assumed to be identical to each other. Weherein consider a more general case where the two waveguidesmay be slightly different in their width and/or refractive-indexincrement. Let K be the coupling coefficient and A the pos-sible mismatch between the propagation constants of the twoguides with no mutual coupling present. According to thecoupled-mode theory,' 8 the conversion length 1, i.e., theminimum length required for complete transfer of light energyfrom one guide to the other in the phase synchronism condi-tion (A = 0), is

1 = 7r/2K, (54)

and the maximum available power transfer F is given by

F = 1/(1 + A2

/4K2

). (55)

On the other hand, in terms of the normal mode analysis suchas presented in Secs. III B and III C, the distributed couplingis interpreted as an interference effect between the symmet-rical and antisymmetrical modes supported simultaneouslyin the coupler region. The propagation constants .3 and i3,of the symmetrical and antisymmetrical modes, respectively,are related to the coupling coefficient K by

K = (,s -fa)/2. (56)

(4

:2:co

X 1 O

g40W

V

o ,

0Z

or

I = (W/a)(O/k)(b - ba')-

A s la/[I(X/h)I = (b )-l.

(57)

(58)

- m=0

-b V=2.0

V=1. 5

111.0

I4T INGPE-WAVEGUJDE MQDE

0 1 2 3 4 5 6NOR. WAVEGUIDE SPACING

7d/a

8

Substituting Eq. (56) into Eq. (54) with use of Eq. (26) yieldsan expression for 1 in terms of the normalized propagation

FIG. 10. Normalized conversion length ,u vs. normalized waveguidespacing dWa for the coupling of fundamental-mode fields.

813 J. Opt. Soc. Am., Vol. 69, No. 6, June 1979

Note that the value of A is uniquely determined for given V,d/a, and m regardless of a. We therefore take this parameteras a normalized coupling length for complete powertransfer.

The normalized conversion length p is plotted in Fig. 10against the normalized waveguide spacing dia for the couplingbetween the fundamental-mode (m = 0) fields guided in V =0.8, 1.0, 1.2, 1.5, and 2.0 waveguides. An important featureof our graded-index device is that the coupling is appreciablystrong, even for rather large waveguide spacing d; and forsmall V, in particular, p increases slowly with d/. Thecoupling characteristics are thus strongly affected by the ex-panse and the amplitude of the evanescent tail of the guidedfield in a guide; the results shown in Fig. 10 are consistent withthe field behavior outlined in the context of Fig. 6. We alsosee that p can be reduced to approximately 10, i.e., a conver-sion length as short as approximately 200 wavelengths is at-tainable for n2 = 2.47 and An2/n20 = 0.01 (a = 0.122). Thepossibility of realizing such a strong coupling leads to relaxedfabrication tolerances, as discussed later. The relation be-tween V and diu which gives constant A is plotted in Fig. 11.From this chart, one can determine the values of the couplerparameters for deriving a prescribed coupling performance.

Figure 12 shows the dependence of A on the x -mode orderm for couplers consisting of V = 2.5, 3.0, 4.0, and 5.0 multi-mode waveguides. We see that, for given V and d/a, an in-crement in m by 1 results in a reduction of p by roughly oneorder of magnitude. This implies that our directional couplerserves as a higher-order-mode-rejection filter with a highextinction ratio. Although the conversion length I dependson the y-mode order n through the j/k factor in Eq. (57), thesensitivity to n is much weaker than that to m.

E. Design examples and discussionsThe practical design of a directional coupler is carried out

by taking the following steps to determine the specific valuesof the device parameters: (i) First give the optical parameters

111 . constants:I I I I I I

lo,

Suhara et al. 813

100

04U)

U)

H

z

a La

mE 102

':2;

z0

H

.40U)

i%rlo,

0z1 . 2

NORMALIZED FREQUENCY V

FIG. 11. Relation between normalized frequency V and normalizedwaveguide spacing dla for obtaining constant normalized conversion length

(n 1 , n2 0, n 3 , and T) of the wave-guiding thin-film structurein which the coupler is fabricated, and determine the effectiveindex of refraction ne for the guided mode to be coupled. For1 = 1.00 (air), n 2 = 2.47 (As 2S2 ) and n3 - 1.54 (#7059glass),

Fig. 4 is used to find ne for a given T. (ii) Decide the value ofthe normalized waveguide width V, which in turn determinesthe relation between the waveguide width 2a and the refrac-tive-index increment An2 via Eqs. (25) and (26). If thewaveguides support many x modes, the possible mode cou-pling may deteriorate the coupler performance. Therefore,it is preferable to select V in the single-mode region given byEq. (35). The specific values of 2o and An 2 are selected ratherarbitrarily under the limitation of the practical fabrication.At this stage, the mode index f/h for the single waveguidemode is found from Fig. 5 and Eq. (26). (iii) Determine thecenter-to-center waveguide spacing 2d for obtaining theprescribed conversion length 1 by finding in Fig. 11 the valueof dl/u corresponding to V and M. (iv) Determine the inter-action length L of the device for realizing a desired transferratio. This can be achieved by simply taking a proper mul-tiple of 1, e.g., L = I for a 0-dB coupler and L = 0.51 for a 3-dBcoupler, etc. Following this procedure, two 0-dB couplers forcoupling of fundamental E& mode field of wavelength A = 1.15pm were designed and the resulting device parameters aresummarized in Table I.

In the case of a conventional step-index coupler, the reali-zation of t as short as 300 pm prescribed in Table I is, if notimpossible, very difficult, because it requires an extremelynarrow lower-index gap and the pronounced sensitivity of I

to small change in the gap width may cause severe problemsin fabrication tolerances. In contrast, specific parameters ofthe graded-index coupler designed herein are quite realizableby the direct electron beam writing technique.

We finally estimate the fabrication tolerances by taking the

0 1 2 3 4NOR. WAVEGUIDE SPACING

5d/vF

FIG. 12. Transverse-mode-order m dependence of the normalized con-version length p.

coupler I described in Table I as an example. Assume thatat least 70 % coupling (F > 0.70) between the two guides is

desired; then we have from Eq. (55) A < 1.31K, which is re-written for A = 1.15 gm and I = 300 gm as

A/k < 0.0013.

Let ba and ba be the mismatch in a and a, respectively, be-tween the two waveguides, as caused by the fabrication error;then we find from Fig. 5 with Eqs. (25) and (26) that themismatch in the propagation constant is expressed for V =

1.5 as

A/k - 0.013(ba/cr) + 0.017(ba/a).

Combining these two relations yields

ba/cr < 0.10 when ba/a = 0,

ba/a < 0.08 when ba/a = 0.

We thus see that 10 % mismatch in the waveguide width or8 % mismatch in the refractive-index increment is allowable.This is an extremely relaxed tolerance in comparison with thestringent condition for a conventional step index coupler.

IV. CONCLUSIONS

We have analyzed graded-index channel waveguides anddirectional couplers fabricated in thin films by the beamwriting technique. The analysis is based on the separationof the wave equation by the effective-index-of-refractionmethod and the solution of the transverse wave equation bythe WKB method. The effective index was determined in away that results in minimum error caused by employing the

TABLE 1. Design examples of directional couplers for coupling of Ex mode fields at A = 1.15,um.

No. ni n2 n3 T(pmj ne V 2ojAmJ An2/n2 O j/k 2d[gm] 1 = L[prm]

COUPLERI 1.00 2.47 1.54 1.50 1.58 0.01 2.28 2.84 300

COUPLERII (air) (As2 S 3 ) (glass) 0.45 2.30 2.00 2.10 0.01 2.29 3.36 500

814 J. Opt. Soc. Am., Vol. 69, No. 6, June 1979

/

,.l1, / / - V=5. 0lI , / , -- V=4. 0

11i ~/ / -v=3.-01 -- V=2.5

I I I I

Suhara et agl. 814

effective index method. It was shown that this techniqueapplies to directional couplers, as well as to channel wave-guides, when special care is taken in connecting the field at-the gap region. It was found that the wave-guiding charac-teristics of a channel waveguide are characterized by a singleparameter V, i.e., the normalized frequency. The guidedmode field has long evanescent tails outside the waveguide andthe positions of the turning points are located further awayfrom the guide center for higher-order modes. Accordingly,the conversion length of a directional coupler increases slowlywith the waveguide spacing and exhibits a strong dependenceon the transverse mode order. It was also found that thestrong coupling attained with a proper choice of the waveguidewidth and spacing results in extremely relaxed fabricationtolerances. The results obtained are summarized in a fewdesign charts by employing normalizations of the device pa-rameters. They are therefore particularly useful for practicaldesign considerations.

By taking advantage of the direct electron-beam writingtechnique, micro waveguides and couplers can be fabricatedfor constructing a highly integrated optical circuit for opticalcommunication and information processing. Experimentalwork is being carried out and the results will be reportedlater.

'E. A. Marcatili, "Dielectric rectangular waveguide and directionalcoupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2102(1969).

2J. E. Goell, "A circular-harmonic computer analysis of rectangulardielectric waveguides," Bell Syst. Tech. J. 48, 2133-2160 (1969).

3J. E. Goell, "Electron-resist fabrication of bends and couplers forintegrated optical circuits," Appl. Opt. 12, 729-736 (1973).

4H. Furuta, H. Noda, and A. Ihaya, "Novel optical waveguides forintegrated optics," Appl. Opt. 13,322-326 (1974).

5M. Sasaki, A. Okamoto, H. Furuta, and T. Kudo, "Directional cou-

plers consisting of modified optical striplines," Technical Digestof International Conference on Integrated Optics and Optical FiberCommunication (IOOC '77), B5-3, Tokyo (1977) (unpublished).

6G. B. Hocker and W. K. Burns, "Mode dispersion in diffused channelwaveguides by the effective index method," Appl. Opt. 16, 113-118(1977).

7H. F. Taylor, "Dispersion characteristics of diffused channel wave-guides," IEEE J. Quantum Electron QE-12, 748-752 (1976).

8T. Taniuchi, M. Kudo, and Y. Mushiake, "Coupling characteristicsbetween optical slab waveguides with gradual distribution of re-fractive index," Trans. IECE of Jpn. J61-C, 256-263 (1978).

9T. Suhara, H. Nishihara, and J. Koyama, "Electron-beam-inducedrefractive-index change of amorphous semiconductors," Jpn. J.Appl. Phys. 14, 1079-1080 (1975).

'0H. Nishihara, "Direct writing technique using a scanning electronmicroscope: fabrication of optical gratings in amorphous chalco-genide films," Technical Digest of International Conference onIntegrated Optics and Optical Fiber Communication (IOOC '77)02-4, Osaka (1977) (unpublished).

"H. Nishihara, Y. Handa, T. Suhara, and J. Koyama, "Direct writingof optical gratings using a scanning electron microscope," Appl. Opt.17, 2342-2345 (1978).

12Y. Handa, T. Suhara, H. Nishihara, and J. Koyama, "Scanning-electron-microscope-written gratings in chalcogenide films foroptical integrated circuits," Appl. Opt. 18, 248-252 (1979).

13W. J. Tomlinson, I. P. Kaminow, E. A. Chandross, R. L. Fork, andW. T. Silfvast, "Photoinduced refractive index increase in poly(methylmethacrylate) and its applications," Appl. Phys. Lett. 16,486-489 (1970).

"4W. J. Tomlinson, H. P. Weber, C. A. Pryde, and E. A. Chandross,"Optical directional couplers and grating couplers using a newhigh-resolution photolocking material," Appl. Phys. Lett. 26,303-306 (1975).

"Integrated optics, edited by T. Tamir (Springer Verlag, Berlin,1975), p. 40.

16L. I. Schiff, Quantum mechanics, (McGraw Hill, New York, 1968),p. 268.

' 7R. Courant and D. Hilbert, Method of mathematical physics(Springer Verlag, Berlin, 1937), chap. 7.

1A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J.Quantum Electron. QE-9, 919-933 (1973).

Assignment of unclassified lines in Tbi throughhigh-resolution laser-fluorescence measurements of hyperfine

structureW. J. ChilIds and L. S. Goodman

Argonne National Laboratory, Argonne, Illinois 60439(Received 7 December 1978)

It is demonstrated that high-resolution measurements of the hyperfine structure of unclassifiedspectral lines can be used in many cases to establish uniquely the classification of the lines if thestructure of the low levels of the atom is already well understood. Examination of nine lines inTh I confirms the previous classification in six cases, requires some modification of the previousassignment in one case, and gives the classification for two previously unassigned lines. One of thenewly classified lines (X5795.64) is the first line found between 4338 and 10 239 A to connect withthe Th i atomic ground state. The Doppler-free laser-atomic-beam technique was used for the study.New hfs results are given for two even levels and eight odd levels of Th i.

The atomic spectrum of Tb i is known to be exceptionallyrich, and to contain many low-lying levels of both parities. Acontinuing series of papers from the Zeeman Laboratoriumby Klinkenberg and Meinders'- 6 has reported great progressin the classification of the enormous number of lines observed.

The 32 lowest, even-parity levels belong to the configuration4f85d6s2 . The levels of 4f85d 26s begin at about 8000 cm-l,and about 100 even levels are now known altogether.7

Klinkenberg and van Kleef8 showed in 1970 that the atomicground state is 4f 6s2 6Hj5/2 , about 285 cm-' below the lowest

815 J. Opt. Soc. Am., Vol. 69, No. 6, June 1979 0030-3941/79/060815-05$00.50 C) 1979 Optical Society of America 815