B? _4n'n? (m+) 7?
moni de Ww’ ,(2)

where bm is the propagation constant for the mth mode and | is the free-space wavelength. This expression can be
further reduced using a paraxial approximation, which gives

g = 20 (m +1)’
2 nN 4nW? (3)

Using this approximation, the propagation constants are separated by multiples of pl/4nW2, and a beat length can
be defined from the propagation constants of the two lowest order modes

Tl

 

 

Oe
74 By te B, (4a)
where, using (3)
2
Oo. 4nW
3h (4b)

In general, when all modes are excited, self imaging will occur at approximately [12]

230
q (Sa)

where p is an integer greater than zero denoting possible device lengths, q is an integer greater than one represent-
ing the number of images, and the two share no common divisor. At L = 3W, the single image is mirrored around the
x axis, while at 6W the self image appears. Using the approximation in (3)

 

_ p4nw*
a ay

L
(Sb)

In the exact analysis, closed-form eigenvalue solutions to Maxwell’s equations are produced. As in the simplest

approximation, an effective index slab approximation to actual devices is invoked. For TE modes, the electric field takes
the form

Aom i"
Ea (A,,, cos(k x)+B,,,sin(k,,,x))e "9; [x|s ¥

xm

am (%F) ,-iBm2g Ww
e HC MEO) Xie

9

y
Wee
em (x+ 2 ) 7 iBm2 e X < -¥

>

2m

(6)

where kxm is the transverse propagation constant in the guide and axm is the decay constant. For symmetric modes
(m = 0, 2, 4, ...), Blm = 0 and for antisymmetric modes (m = 1, 3, 5, ...), Alm = 0. In the exact case the propagation
constants of all modes are calculated from the analytic determinantal equation using a hybrid Newton-Raphson and
bisection routine with derivatives. Using these propagation constants and calculating coupling through input and out-
put guides using overlap integrals the output fields in each output guide are obtained. |

The exact propagation constants obtained from the determinantal equation may be used in (4a) to formulate a sec-
ond approximation of the length of the MMI section.

 

RESULTS

Cascaded devices consisting of an NxN coupler connected with electro-optic phase shifters to a reversed 1xN split-
ter are considered. The phase shifters are assumed to be continuously tunable. Each device is modeled as an
InP/InGaAsP/InP deeply etched rib waveguide structure with an effective guiding index of 3.25 and a substrate index
of 3.20; device optimization is performed at | = 1.319 jm. The effective index outside the guiding region is taken as one
since the rib is assumed to be etched through the guiding region. A 2x2x1 cascaded device is considered. All input, out-